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Algorithmica (2001) 29:148-180 DOI: 10.1007/s004530010059 Algorithmica

~) 2001 Springer-Vcrlag New York Inc.

Near-Optimal Bounded-Degree Spanning Trees

J. C. Hansen 1 and E. Schmutz 2

Abstract. Random costs C(i, j) are assigned to the arcs of a complete directed graph on n labeled vertices. Given the cost matrix Cn = (C(i, j)), let Tt* = T~(Cn) be the spanning tree that has minimum cost among spanning trees with in-degree less than or equal to k. Since it is NP-hard to find Tt*, we instead consider an efficient algorithm that finds a near-optimal spanning tree Tff. If the edge costs are independent, with a common exponential(1) distribution, then, as n ~ oo,

E(Cost(Tff)) = E(Cost(T~)) + o(i). Upper and lower bounds for E(Cost(T~)) are also obtained for k > 2.

Key Words. Approximation algorithm, Spanning tree, Assignment problem, Random mapping.

1. I n t roduc t i on . In this paper random costs are assigned to the edges of the complete directed graph, and the expected cost of the cheapest k-tree is estimated. (By "k-tree" we mean an in-directed spanning tree whose maximum in-degree is at most k.) Let Cn = (C(i, j ) ) denote an n x n matrix whose entries are independent exp(1) random variables. The variable C(i, j ) is understood to be the cost of the directed edge, (i, j ) , from vertex i to vertex j in the complete directed graph, Dn, with vertices labeled 1, 2 . . . . . n. Note that loops (i, i) are included in D~ (so Dn has n 2 directed edges).

It is an NP-hard problem to find an optimal k-tree given a cost matrix Cn as input (Lemma A. 1 in the Appendix). Since it is hard to find an optimal k-tree, we propose a heuristic algorithm. A near-optimal bounded-degree spanning tree can, with high probability, be obtained by easy modifications of the cheapest k-map. (A k-map is a subgraph of Dn such that each vertex in the subgraph has out-degree 1 and has in-degree less than or equal to k.) This is significant because it is computat ional ly easy to find the cheapest k-map. To analyze the algorithm, we prove that the average cost of the optimal k-tree is asymptotical ly close to the average cost of the cheapest k-map. We also provide upper and lower bounds for the expected cost of the optimal k-map (and, hence, bounds for the expected cost of the optimal k-tree) for all k > 2.

We note that, for k -- 1, a k-map is in fact a permutation. So our problem is related to the assignment problem. Our methods for obtaining a lower bound for the expected cost of the optimal k-map are different from those of Goemans and Kodialam [ 1 ] and Olin [2] who compute the expected value of a feasible solution to the dual l inear program for the assignment problem. We believe their methods yie ld better bounds than ours in the case

1 Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, Scotland. J.Hansen @ ma.hw.ac.uk. 2 Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA 19104, USA. eschmutz@ mcs.drexel.edu.

Received September 15, 1998; revised March 16, 1999. Communicated by H. Prodinger and W. Szpankowski. Online publication October 6, 2000.

Near-Optimal Bounded-Degree Spanning Trees 149

k --- 1, but that our methods give better results for k > 2. Coppersmith and Sorkin [3] have recently made significant progress on the upper bound for the assignment problem. Their methods are appealing, and may be relevant here, but we have not been able to exploit them directly. The work of Frieze et al. [4]-[6] deals with analagous problems for undirected graphs. There is also literature dealing with bounded degree spanning trees in the plane and other "geometric" analogues (e.g. [7]). These problems are only superficially similar to our problem.

To obtain an upper bound on the expected cost of the optimal k-tree, we bound the expected cost o f a k-map that is constructed using a greedy heuristic algorithm. Although this greedy heuristic is not optimal, it yields a map whose expected cost is bounded and surprisingly close to the optimum. It is interesting to compare this with greedy heuristics for the assignment problem which yield very poor assignments having ® (log n) expected cost. As a referee pointed out, this is because the case k = 1 is more constrained: when edges are added using a greedy heuristic, the number of potential edges decreases steadily so that the last few edges contribute significantly to the expected cost. This is less of a problem for k > 2 because there is more flexibility in selecting edges.

Finally, we remark that the case k ---- oo is the problem of finding the optimal spanning tree with no degree restrictions. In this case, greedy heuristics work well asymptotically. In particular, greedy methods have been used to obtain limit theorems for the expected cost of the optimal spanning tree, and for the distribution of the cost of the optimal tree as n ~ oo [8], [9]. Thus, when estimating the expected cost o f the optimal k-tree, k -- 1 is the difficult case, k ----- oo is the easy case, and 2 < k < oo is intermediate in difficulty.

A little notation is needed to proceed. For 1 < i < n, let C~l) (i), c~2) (i) . . . . . c~n) (i) denote the order statistics of the variables {C(i, j ) : 1 < j < n }. The joint distribution o f the order statistics of i.i.d, exponential random variables is well understood. We make use of the fact that c<k) (i) ",~ Rn + R n - i "JF''" "1- R~_k+l where [ Rm, 1 < m < n } are independent random variables with Rm "~ exp(m). It is also a consequence of the "memoryless" prop- erty of the exponential distribution that c~l) (i), c~2) (i) - c<t) (i) . . . . . cc~ ) (i) - cc,_ L) (i) are independent with c~k)(i) - CCk-l)(i) "~ exp(n -- k + 1) for 1 < k < n. Finally, for 1 < i < n and any vertex v, define X<i)(v) = j if and only if C ( v , j ) = c~i)(v). For each vertex v, the vector (Xo)(v) , X~2)(v) . . . . . Xtn) (v ) ) is a uniform random permutation of the vertices 1, 2 . . . . . n. One can also verify that for each vertex v, the variables {Xti)(v): 1 < i < n} and {c<i)(v): 1 < i < n} are independent. It follows that the a-algebras a{X(i)(v): 1 < i < n, 1 < v < n} and tr{c<i)(v): 1 < i < n, 1 < v < n} are independent too.

Given Cn, let Tk* = T~(C~) be the cheapest k-tree, and let M~ = M~(Cn) be the cheapest k-map. It is helpful to think of M~ as a (nonuniform) random map; we write M~ (v) = w iff (v, w) ~ M~. Hansen [8] observed that M~* is in some sense close to being a minimum spanning tree in Dn: by breaking a few cycles in M~* and redirecting some edges one can obtain a spanning tree whose expected cost is asymptotically optimal. A similar strategy is developed here for bounded-degree spanning trees. The idea is very simple. First create a forest by removing one edge f rom each cycle of M~. Then patch together the components of the forest to form a tree. I f r is the root of a tree in a forest, call v avai lable f o r r if the in-degree of v is less than k and v is not in the same weak component as r. I f we adjoin the edge from r to v, the result is a forest with one less

150 J .C . Hansen and E. Schmutz

component. This is the basis for

Algorithm 1

1. Find M~. 2. Let Tk a be the forest obtained by deleting the most expensive edge from every

cycle of M~, and let ~¢ be the number of components that Tk a has. 3. F o r / = 1 . . . . . r - 1

{ Let r i be the root of the smallest component of Tk a. Add to Tff the cheapest edge in/9, from ri to a vertex that is available to ri. }

Algorithm 1 creates a k-tree T~ in polynomial time. To see this consider the following linear program L P (C,, k):

Minimize Z = ~ ~ C( i , j ) x i , j i = l j = l

Subject to:

~-~Xi, j << k ( j = 1, 2 . . . . . n), i=1

~ Xi, j = 1 (i = 1,2 . . . . . n), j = l

xi,j > 0 (1 < i , j < n ) .

Any 0-1 feasible solution to this LP corresponds to a k-map M. The correspondence i s Xi, j = 1 if (i, j ) e M, and Xi, j = 0 otherwise. The first n constraints say that each vertex has in-degree less than or equal to k, and the second n constraints say that each vertex has out-degree one. It is a well-known theorem in linear programming that the optimal solution to this kind of transportation problem is in fact an integral solution, and so the optimal solution to the linear program L P (C,, k) is a 0-1 solution [ 10]. Since M~ corresponds to the optimal solution to the linear program, the first step in Algorithm 1 can be solved in polynomial time and the remaining steps can also be carded out in polynomial time.

In this paper we prove that Algorithm 1 is asymptotically optimal: in Section 2 we show that E(Cos t (T: ) ) = E(Cost(T~)) + o(1). In Section 3 we obtain a lower bound for E(Cos t (T: ) ) and in Section 4 we obtain an upper bound by analyzing a "greedy" algorithm.

2. Analysis of Algorithm 1. The main goal of this section is to prove

THEOREM 2.1. Ifk >_ 2, then E(Cost(T~)) = E(Cost(T~)) + o(1).

We establish Theorem 2.1 by showing that E(Cost(M~)) is close to both E(Cost (T~)) and E (Cost(Tk*)). The argument is similar to Karp and Steele's [ 11] analysis of a patching algorithm for the asymmetric traveling salesman problem. The first step is to prove

THEOREM 2.2. l f k >_ 2, then E(Cost(T~)) = E(Cost(M~)) + o(1).

PROOF. Fixk > 2 and define the subgraph D'~ of Dn as follows:edge (i, j ) ~ D~ if and only if C(i, j ) < L(n) where L(n) = (50 log 2 n) /n . Let M~ denote the cheapest k-map in D~,, provided such a map exists. (M~ does not exist if, for example, there is a vertex

i such that (i, j ) ¢ D', for every 1 < j < n.) Let/(/k = M~ if M~ exists; otherwise, let /17/k = M~ and consider the following modification of Algorithm 1.

Algorithm 1'

1. Find Mk. 2. Let 1"~ be the forest obtained by deleting the most expensive edge from every

cycle of/~k, and let t~ be the number of components that T~ has. 3. Fori = 1 . . . . . k - 1

[ Let ri be the root of the smallest component of Tff.

Add to I"~ the cheapest edge in Dn from ri to a vertex that is available to ri. }

Now let Bn = {)~/k = M~,t~ < log 2n, Fi < log 4 n , i ----- 1,2 . . . . . n} where Fi ----- I{J: C(i, j ) < L(n)}l. Given B~, the combined cost of the edges deleted in Step 2 of Algorithm 1' is at most t~ • L(n) < (50 log 4 n) /n . Hence,

E(Cost(l(Ik)lB,) 501°gan < E(Cost(f '~)lBn) n

< E(Cost(lt4k)[B,) + E(Cost(added edges)lBn).

We show below that E(Cost(added edges)lB,) < (100 log 4 n) /n by bounding the expected cost of each edge added by Algorithm 1'.

For i < ~ < log 2 n, consider the ith iteration of Step 3 in Algorithm 1'. Let .Ai --- .Ai (ri) denote the set of vertices that are available to ri at the beginning of the ith iteration of Step 3. The edges out ofri are examined in increasing order of cost until an edge that points to a vertex vi ~ .Ai is found, then the edge (ri, vi) is added to Tff and the added cost is C(ri, Ui). Thus,

E(Cost(addededges)lB~) = E C(ri, 1)i) Bn i = I

log 2 n

----- Z E(C(ri , vi)li < k, Bn)Pr(i < k l B , ) i=1

log 2 n

< ~_, E(C(ri , vi)li < ~, Bn). i=1

To bound E(C(ri , Vi)li < K, Bn) for each i < log2n, we analyze a more expensive "patching" operation which is described below.

Fix i < log2 n. Given i < k and Bn, letmi = n - Fr, = I{J: C(ri, j ) > L(n)}l, and let .A~ = {j ~ .Ai : C (ri, j ) > L (n)}. Call any edge (ri, j ) with C (ri, j ) > L (n) a costly

152 J .C. Hansen and E. Schmutz

edge. Now modify the ith iteration of the patching operation in Step 3 of Algorithm 1' as follows: add to ]'k a the cheapest edge from ri to a vertex in .4 I. In other words, at the ith iteration of the algorithm we examine only costly edges out of ri in increasing order of cost until we encounter one that points to a vertex wi ~ .4'i and edge (ri, wi) is the

new edge that is added to 7~k a. Observe that C(ri, vi) < C(ri, wi) always, where vi is the cheapest vertex available to ri in .4i. Thus any upper bound for E(C(r i , wi)li < ~, Bn) is also an upper bound for E(C(ri , vi)li < ~, Bn).

To bound E(C(ri , wi)li < ~, Bn), observe that given that the edge C(ri, w) > L(n), we have C(ri, w) "~ L(n) + X, where X --- exp(1). (We use the fact that if M~ exists, then a "cheap" k-map has been constructed without examining any of the costly edges, so we have no extra information about the costly edges.) It follows from standard results for order statistics of exponential random variables that if there are mi costly edges out of vertex ri and if the dth cheapest costly edge out of ri is added to ]'k a, then

1 1 1 E(C(r i , wi) [ d, mi, i < ~c, Bn) -~ L(n) + ~ + ~ + . . . +

mi rn i -- 1 mi -- d + 1"

The random variable d has the same distribution as the number of draws, without replacement, until a black ball is drawn from an urn with 1.4~1 black balls and mi - 1.4~1 white balls. In Lemma 2.9 we prove that 1.4il > n/4, so 1.4'il >- 1.4Jl - Fr, >_ n / 4 - 1 o g 4 n > n/5 for all large n. Hence

P r ( d > 5 1 o g n l m i , i < ; c , B n ) < ( 1 1"411) 51°gn ( ~ ) 51°g~ 1 _ - < < - n n

for all large n. Also, given B~, we have mi > n - log 4 n and thus

E(C(ri , vi)li < ~, B~) < E(C(ri , wi)[i < ;c, B,) 5 log n

< L(n) + mi - 5 log n - log 4 n

+ ( ~ _ _ ~ ) P r ( d > 5 1 o g n l i < ~ , B n ) < 2 L ( n )

for all sufficiently large n. Hence

log 2 n

E(Cost(added edges)lB,) < E E(C(ri , vi)li < ~:, B~) i--1

< 2L(n) log2n = 1001°g 4n n

In Lemma 2.8 we prove that Pr(B c) = O(1/nS), so it follows that for all large n,

E(Cost(iVlk)) - E(Cost(T~))

150 log4 n IE((Cost(l(4k) Cost(7"ff))l{BnC}) < n + - "

1501°g4n ( ) < + E(Cost(J(tk)2) 1/2 + E(Cost(7"~)2) 1/2 (Pr(BnC)) 1/2.

n

Cost(I~lk) < ~_,in=l C(m(i) always, so we have

E(Cost(lVIk) 2) < E c(n~(i = Var c~,)(i + E c~n)(i

n l n2 ( ~ 1) 2 = n Var(c(n)(1)) -t- n2(E(c(n)(1))) 2 = n Z ~ A-

k=l k=l < 2n 2 log 2 n

since C(n)(1) "~ g l + R2 + "'" + Rn. Similarly, E(Cost(Tr~)2) < 2n 2 log 2 n. Thus

[ E(Cost(1~tk)) - E(Costd'~)) < 200 log 4 n n

To finish the proof, we note that whenever )Qk = M~ we must have 7~k a = Tff too. Now it

follows from Lemma 2.4 below that Pr(AT/k = M~) > Pr(AT/k = M~ = M~) = 1 - O ( ~ ) , and so by arguments similar to those given above, we have

[E(Cost(M~) - Cost(T~))[

< 2501°g4n + E( (Cos t (M~) -Cos t (T~) ) . l{)~/k # M~}) - n

300 log 4 n < n

and except for the unproved lemmas that were cited, we have now completed the proof of the theorem. []

The proof of Theorem 2.2 used several lemmas that must now be proved. In particular, a key step in the proof of Theorem 2.2 is the observation that M~ exists and equals M~ with high probability. To establish this we modify an argument from [11]. We begin by defining the directed subgraph a(Cn) of Dn in which

(i, j ) is an edge of G(Cn) 1 6 l o g n

¢> C(i, j) < p(n) -- - - n

For any subset S _ [n], define I-'(S) = {j: (i, j) ~ G(Cn) for some i 6 S] and F-1(S) = {i: (i, j) ~ G(Cn) for some j 6 S}. The directed graph G(Cn) is called expanding if, for any subset of vertices S _ [n], the following inequalities both hold:

[F(S)I > m i n { 2 l S l + l , n - - ~ l / and [ F - l ( s ) [ > m i n { 2 l S l + l , n - - ~ l I .

Then we have

LEMMA 2.3. Pr(G(C,) is not expanding) = O(1/nS).

154 J.c. Hansen and E. Schmutz

PROOF. This lemma is essentially Lemma 7 o f [11]. The only difference is that Karp and Steele use uniformly distributed cost variables and set p(n) = 10 log n / n to obtain a probability bound which is O (1/n2). Their proof goes through with trivial modifications when the cost variables are exponential(l) and p(n) = 16 log n /n , so we do not repeat the argument here. []

LEMMA 2.4. With probability 1 - O (1/nS), every edge of M; has cost less than L (n) = (50 log 2 n) /n .

PROOF. Observe that by Lemma 2.3, it is enough to prove that if G(Cn) is expanding, then every edge of M; has cost less than (50 log 2 n) /n . So suppose G(Cn) is expanding but C(i ' , M;( i ' ) ) > (50log 2 n ) / n for some 1 < i ' < n. We show that the mapping M ; can be modified to obtain a feasible solution Mk which is cheaper than M~.

The first step is to define a sequence of subsets of vertices as follows. Let F(1) = 1"({i'}) and for I > 2, let F(/) = F ( ( M ; ) - I ( F ( I - 1))). Let .A(M;) denote the set of vertices in M; with in-degree less than k. Since each vertex has in-degree at most k under M~, we must have IA(M~)I >_ [n/21. We claim that {/: F(I) O.A(M~) ~ 0} # 13. To see this, note that if IF(/)I > (n + 1)/2 for some l, then F(/) f3 .A(M~) ~ 0 (since IA(M;)J > [n/2]) . So if {I: F(I) N .A(M;) ¢ 13} = 13, then ]F(I)I < (n + 1)/2 for all l > 1. On the other hand, since G(Cn) is expanding, IF(l)[ _> min(2, (n + 1)/2), and since 1-'(1) fq .A(M~) = 0, every vertex in F(1) must have in-degree k under M; . It follows that I (M~)- l ( I ' (1))[ > 2k and

1I"(2)1 = tF ( (M; ) - I (F (1 ) ) ) [ > min(22k, (n + 1)/2).

Now induction shows that IF(I)I > min(21k t - l , (n + 1)/2) for all l > 1, and so IF(l)[ > (n + 1)/2 for l > 21ogn. Thus we cannot have IF(I)I < (n + 1)/2 for all l > 1.

Let m = min{e: r'(e) n .A(M;) ¢ 13} and note that it follows from the argument above that m < 2 log n. The next step is to define two sequences of vertices il, i2 . . . . . im and j l , Jz . . . . . jm. Let il = i ' and let jm be a vertex in F(m) M .A(M~) ¢ 0. Since jm 6 F(m) fq .A(M;) there is a vertex i,n 6 ( M ; ) - l ( F ( m - 1)) such that C(im, jm) < p(n) = (16 log n) /n . The remaining vertices in the sequence are defined recursively as follows. For 1 < l < m 1, let jt * " ~ - = Mk(tt+~) F(/) = I - ' ( (M~)-I(F( / 1))). For 2 < l < m - 1, choose it ~ (M~) - l (F(I - 1)) such that C(it, jl) < p(n) . Observe that jm ~ * ' = J t f o r l < m - * = M k (tt+l) l < 1 since M;(it+l) ~ F(/) and F(l) fq A(M~) ~3 f o r l < l < m - 1 .

Given the two sequences il, i2 . . . . . in and j l , j2 . . . . . Jm, define a new mapping Mk by setting Mk(i) = M;( i ) i f / ¢ {il, i2 . . . . . ira} and setting Mk(it) = jt for 1 < l < n. In other words, Mk is constructed from the optimal mapping M~ by deleting the edges (it, M;( i t ) ) and adding the edges (it, jl) for 1 < l < n. To see that Mk is a feasible solution, we note that for each 2 < l < m, the deletion of edge (it, M~ (it)) makes vertex M;(i t ) "available" and so the addtion of the edge (it-l, j l - l) = (it-l, M;(il)) does not violate the degree constraint at v e r t e x M;(il). Also, since jm ~ jt for 1 < l < m - 1, the vertex j,, is still "available" after the addition of the edges {(it, jl): 1 < l < m - 1}, so the addition of edge (in, jm) does not violate the degree constraint at vertex jm. So the mapping Mk is a feasible solution.

Finally, observe that

m £

Cost(Mk) - Cost(M;) = ~ C(i,, jr) - C(it, M;(i , )) t=l t=l m

<_ ~ c ( i , , j,) - c(i~, M~(il)) t=l

50 log 2 n < m • p(n)

n

50 log 2 n _< 21ogn • p(n)

n

which contradicts the optimality of M;.

- 1 8 l o g 2 n

n

[]

Next we establish that, with high probabilty, M~ has less than log 2 n components. It is well known that, for uniform random maps, the number of components is O (log n) with high probabilty, and it would not be difficult to prove the same fact for uniform random k-maps. However M; is a nonuniform random k-map. Nevertheless, the corresponding statement is a consequence of the following

LEMMA 2.5. Let f and g be two k-maps that differ only by a transposition of the values they assign to two vertices, i.e. there exist il, i2 such that f (il) = g(i2), f ( i2) = g(il), and, for v ~ il, i2, f (v) = g(v). Then

P r ( f is optimal) = Pr(g is optimal).

PROOE Let C be the set of cost matrices, and for any k-map M, let O~ c C be the set of cost matrices for which M is optimal. We want to prove that Pr(Of) = Pr(Og). Define H: C ~ C by H(C) = C', where

C(i2, j ) if i = iz, C ' ( i , j ) = C ( i l , j ) if i = i 2 ,

C(i, j ) else.

We know Pr(Of) = Pr(H(Of)) because costs are assumed i.i.d. It therefore suffices to prove that H ( O f ) ---- Og.

Let C ~ Of. To prove that H ( O f ) c Og, it suffices to show that g is optimal for the instance C'. First note that Cost(f, C) = Cost(g, C'):

Cost(f, C) = C(il, f ( i l ) ) + C(i2, f( i2)) + ~ C(i, f ( i ) ) i~il ,i2

= C(il, g(i2)) + C(i2, g(il)) + ~ C(i, g(i)) i~it ,i2

= C'(i2, g(i2)) + C'(il, g(il)) + ~ . . C'(i, g(i)) i~il ,i2

= Cost(g, C').

Now we prove by contradiction that g is optimal for C'. Suppose, on the contrary, that h is a k-map and Cost(h, C') < Cost(g, C'). Define h ' by h ' ( i l ) = h(i2), h'(i2) = h( i t ) , and, for i ~ il, i2, h'(i) = h(i). Then Cost(h', C) = Cost(h, C') < Cost(g, C') = Cost(f, C). This contradicts the optimality of f , and completes the proof that

(2.1)

By the same argument,

(2.2)

H ( O f ) _ Og.

H(Og) c_ Of.

Observe that H 2 is the identity. Hence, by applying H to (2.2) we get

(2.3) Og c H(Of) .

Combining (2.1) and (2.3) we get H(Of) = Og. []

For any map M, let Z = Z(M) = {i: Mt(i) = i for some t > 1} be the set o f cyclic vertices of M. Then M[z is a permutation on Z. The following corollary asserts that, given the set Z of cyclic vertices, all permutations are equally likely to occur as M~ I z.

COROLLARY 2.6. For any set Z c_ {1, 2 . . . . n} and any permutation ¢r on Z,

1 Pr (M~lz = t r lZ(M~) = Z ) = IZl!"

PROOF. m transpositions such that zr = ~r o l-[i=l ~i. By Lemma 2.5, we have

For any two permutations Jr, tr of Z , there is a sequence r l , l"2, . . . , ~'rn of

f o r l < g < m . []

LEMMA 2.7. With probabilty 1 + o(1/ nS), M~ has less than log2n components.

PROOF. Let m _< n be the number of cyclic vertices, and let r be the number of cycles. By Corollary 2.6, we need only estimate the probability that a uniform random permutation on m letters has more than log 2 n cycles. It is well known (e.g. [12]) that this probability is negligible: there is a constant C > 0 and a positive constant a < 1 such that for all t > 0,

I ) P r \ ~ > t I Z ( M D I = m < C u t .

Take t = log 4/3 n to obtain the result. []

LEMMA 2.8. Pr(Bn) = 1 -- O(l/n5).

PROOF. Recall that Bn = {/~/k = M~, k < log 2 n, F/ < log 4 n, i = 1, 2 . . . . . n} where Fi = [{j: C(i, j) < L(n)}[. Therefore,

n

Pr(B~) < Pr(,Qk ~ M~) + Pr(k > log 2 n) + ~ Pr(Fi > tog 4 n). i= l

Now if every edge of M~ has cost less than L(n) = (50log 2 n)/n, then M~ must exist and M~ = M~. So it follows from Lemma 2.4 that

Pr(l~lk=M'k)>Pr(J~Ik---M'k=MD> l - - O ( ~ 5 ) .

T h u s

Pr(B~) _<

<

Pr(~ >log:n, Mk =M'k = M ~ ) + O ( ~ ) + ~--~Pr(Fi > log4n) i=1

Pr(K>_log2n)-t-nPr(Fl > log4n) q- O ( ~ 5 ) •

By Lemma 2.7, Pr ( r > log 2 n) = o(1/nS). Finally, since F1 ~ Bin(n, 1 - e x p ( - L ( n ) ) ) ,

E(eF,) e50(e-1)log 2n (_~ ) Pr(FI > log4n) = Pr(e Ft > n l°g3n) < - - < -- o . []

- - n l ° g 3 n - - n l ° g a n

Lastly, we prove

LEMMA 2.9. For 1 n/4.

PROOF. Let .T~,. be the forest that consists of all components other than that of ri. Let dj (.T'i) denote the number of vertices in ~ . having in-degree j . The number of available vertices is

k -1

Iail = ~_,dj(.Tci). j=0

The number of vertices in .,~,. is

k

v(i) = ~ dj(.,~i). j=O

Since .~i has v(i) vertices and ~ - i components the number of edges of .T/is

k

v(i) - ~ + i = ~j-~jaj(.~,.). j=l

It follows that

0 < ~ - i = v ( i ) - ~ j d j ( ~ ) < v ( i ) - e l ( ~ ) - 2 ai(.~,.) j=t j=2

= -v( i) + 2do()V, .) + d l (~ / ) .

158 J . C . Hansen and E. Schmutz

Hence

2(d0(Sr,, ) + dl (.~']i)) > v(i).

Provided k > 2, we have Iail > (d0(.T/) + dl(.T/)). We know v(i) > n / 2 because ri was the root of the smallest component. Therefore IAi[ > n/4 . []

Now that the lemmas are proved, the proof of Theorem 2.2 is complete. However, our main aim is to prove

THEOREM 2.1. l f k > 2, then E(Cost(T~)) = E(Cost(T~)) + o(1).

PROOF. Let r be the root of Tk*, and let (r, w) be the cheapest edge from r to a vertex having in-degree less than k in Tk*. Then

and consequently

Cost(M~) < Cost(T~) + Cost((r, w))

E(Cost(T~)) >_ E(Cost(M~)) - E(Cost((r, w)).

We claim that E(Cost((r, w))) < (21og4n) /n for all large n. To prove this we show that C(r, w) is bounded above by a random variable Y whose expected value is bounded by (21og4n)/n. To define Y we introduce some notation. For each 1 < i _< n, let Tk(i) be the cheapest k-tree rooted at vertex i, and let (i, wi) denote the cheapest edge from i to a vertex having in-degree less than k in Tk(i). Note that the variables Cost(l, Wl), Cost(2, w2) . . . . . Cost(n, Wn) are identically distributed, though not independent. Now let Y = maxl<i<, Cost(i, wi). Since T~* = Tk(i) and (r, w) = (i, wi) for some 1 n

,og'n ( ( <_ ~ + (E(y2) ) 1/2 Pr Y >

n n

log 4 n Io n <_ ~ + (E(y2) ) 1/2 Pr Cost(i, wi) >

n

log4 ( ( < ~ + ( E ( y 2 ) ) 1/2 nPr Cost(1, wl)>_

n n

To b o u n d ( E ( y 2 ) ) l / z , le t Z = m a x{C( / , j ) : 1 < i < n, 1 _< j < n} and note that (E(y2) ) 1/2 < E ( Z 2 ) 1/2. The variable Z has density f (z) = n2e-Z(1 - e-Z) "2-a, so

( E ( y 2 ) ) l / 2 < ( n 2 f o ° ° Z 2 e - ~ ( 1 - e - Z ) " 2 - ' d z ) a / 2

< n 2 z2e -z dz = ~/-2n.

Next, let W = I{j: C(1, j ) < (log 4 n)/n}l and let U denote the random set o f vertices in Tk(1) with in-degree equal to k. Since W' := n - W "- Bin(n, exp(-( log4n)/n)) , we have

el°g2n E(e w') C (2.5) Pr(W < log2n) = Pr(W' > n - log2n) < e" <e log " - ' - -~

for some constant C which does not depend on n. Also, since at most n / k vertices in Tk (1) can have in-degree k, lUl <_ n / k always. For 1 < i < n, let Xfi) = X(i)(1) denote the vertex to which the ith cheapest edge out o f vertex 1 points, then (Xo) , X(2 ) . . . . , Xfn)) is a uniformly distributed random permutation of the set { 1, 2 . . . . . n }. The variables Xf~), X(2), • • •, Xfn) are measurable with respect to the a -a lgebra generated by the variables {C(1, j ) : 1 < j < n}, whereas the random tree Tk(1), and hence the random set of unavailable vertices L/, is determined by the variables {C(i, j ) : 2 b and the set b/, we have Cost(l, wl) > (log4n)/n only if X(1), Xf2), • • . , Xfb) ~/, / , i.e. we must reject at least the first b vertices that are examined and this happens only if all of those vertices are in/.4. So

( l°:n I ) Pr Cost(l, Wl) > W > b , L / = U t7

< Pr (X(t), X(2) . . . . . X(b) ~ UIW > b, lA = U)

( ) U I - . 1 ( [ V ] - b + l ) \ .(1:(, since IUI _< n/k . Using this bound, we obtain

( l°g4 n ) (2.6) Pr Cost(l, wl) >

n

_ < Z P r Cost(1, wx) >_ W > _ b , U = U u n

x Pr(W >_ b,/X = U) + Pr(W _< b)

< 1 - Pr(W > b) + Pr(W < b),

where the sum is over all possible set values for the random set/,/. It follows f rom (2.5) and (2.6) with b ----- log 2 n, that

( log4 n ) C ' Pr Cost(l, wx) > <

- n - n (l°g2)(l°gn) '

where C ' is a constant which may depend on k but which does not depend on n. Using this bound and the bound for (E(y2))1/2 in (2.4) we obtain

2 log 4 n E(Cost(r, w)) < E(Y) <

n

for all large n.

160 J . C . H a n s e n and E. Schmutz

Finally, since Cost(T~) < Cost(T~) by the definition of Tk*, and since E(Cost(T~)) = E(Cost(M;)) + o(1) by Theorem 2.2, we have, for all large n,

2 log 4 n E(Cost(M;)) - - < E(Cost(T~)) < E(Cost(T~))

n = E(Cost(M~)) + o(1). []

3. Lower Bound. In this section we prove the following lower bound for the optimal tree's expected cost:

THEOREM 3.1. If k >_ 2, then

1 1 ~ m - k - £ + l liminfE(Cost(T~)) > 1 + - n ~ - e m . l e = 1 m - £ + l

PROOE By Theorems 2.1 and 2.2, it suffices to prove that

1 1 ~ . 7 - ~ m - k - e + 1 liminfE(Cost(M;))nooo > 1 + e ~ ~. . ~e=l t~/Z'~ -~ i

Since the optimal mapping M~ must have one edge out of each vertex, a crude lower bound is

Cost(M;) > ~ cO)(v) = Cost(M*), I)

where cti)(v) denotes the cost of the ith cheapest edge out of v, and M* = M~* is the cheapest map (with no restrictions on the in-degrees of the vertices). With high probability, a positive proportion of the vertices of M* have in-degree larger than k. Hence a positive proportion of the edges in M* cannot be used in M; . Each time the cheapest edge is not used, we pay a penalty, and the idea is to estimate this penalty.

First some notation is needed. For any vertex v and 1 y ~ c~2)(v) - co)(v) = ~ D(v). v s.t. el(v)¢M; v s.t. el(v)¢M;

We interpret D(v) as the penalty which is paid for not using edge e~(v). For any map f from {1, 2 . . . . . n} into {1, 2 . . . . . n} and any vertex o, let p(o, f ) denote the in- degree of v in G f , the directed graph representing the mapping f . For 0 < m < n, let Vm(f) := {v: p(v, f ) = m} and for any vertex y, let B(y, f ) := {v: f ( v ) = y}. In the special case where f = M*, we write Vm for Vm (M*) and 13(y) for /3(y , M*).

Now given y ~ Vm and B(y) = {xl, x2 . . . . . Xm}, let Do)(13(y)), D(2)(B(y)) . . . . . denote the order statistics of the variables D(xl), D(x2), . . . , D(xm) . For each vertex y, we define X(y) = 0 if y ¢ );m for some m > k and X(y) = ~,m~k D(i)(13(y)) if Y ~ ))m for some m > k. The variable X (y) is a lower bound on the cost of redirecting

edges that go into vertex y under M*. In particular, if y ¢ )),, for some m > k, then it is possible that none of the edges into y under M* are redirected under the optimal solution M; , whereas, if y e ))m for some m > k, then at least m - k o f the edges into y (under M*) must be redirected under M ; and the cost of redirecting edges will be at

least ~m=~k D(i)(]3(y))" Using the variables X(1), X(2) . . . . . X(n), we define a variable Un that underestimates the cost of M~:

rn-k n Cost(M;) > Un : : Cost(M*)-4- ~ ~ ~ D(i)(13(y)) : Cost(M*)-4- ~ X(y) .

m>k yEVm i=1 y=l

The variable Un is the sum of the cost of the cheapest edge out of each vertex, plus an additional penalty for vertices y with in-degree m > k. The variables X(1), X(2), . . . , X(n) are identically distributed, so

E(Un) = E(Cost(M*)) + nE(X(1)) = 1 + n E ( X (1))

since E(Cost(M*)) = hE(c{1}(1)) = 1. Letpm(n) = Pr(1 E ))m) = Pr(p(1, M*) : m) = (n)(1/n)m(1-1/n)n-m "~ 1/em!,

then we have

E(X(1) ) = ~ E(X(1)I1 E ))re)pro(n). m>k

To compute E(X(1)I1 ~ 13m) we write

E(X(1)I1 e Vm) = ~_E(X(1) IB(1) = .4) Pr(B(1) = .411 ~ Vm), A

where the sum is over all subsets .4 ___ {1, 2 . . . . . n} such that 1.41 = m. Recall that the a-algebrasa{X(i)(v) : 1 < i < n , 1 < v < n } a n d t r { c ( i ) ( v ) : 1 < i < n , 1 < v < n } are independent. Now observe that given B(1) = .4, then X(1) = )--~m~ D(O(.4). For any subset .4, the event {B(1) = .4} is measurable with respect to tT{X(i)(v): 1 < i < n, 1 < v < n} whereas the variable X(1) = )--~m~k D(i}(.4) is measurable with respect to ~r{c(i)(v) : 1 < i < n, 1 < v < n}. So by independence o f the a-algebras, we have

E(X(1)IB(1) = .4) = E /3(1) = .4 = E(D(i)(.4)) *= i=l

for any subset "4 ~ {1, 2 . . . . . n} such that 1"41 : m. Also, for any subset .4 { 1 , 2 , . . . , n } we have

m-k m-k

E(O(i) (~4)) = ~ E(O(i)(~4')) , i=l i=1

where "4' = {1, 2 . . . . . m}, since the variables {D(i): 1 < i < m} and {D(x): x ~ "4} have the same joint distribution. Since the variables D(1), D(2) . . . . . D(m) are i.i.d.

162 J.C. Hansen and E. Schmutz

exp(n - 1) random variables, it follows from Lemma 3.2 below that

m - k

E(X(1) ) I ~ "Vm) = E E E(D(i)( .A'))Pr(B(1)= ,411 ~ V.,) .A i=1

m - k

= E E(D(i)(A')) i=1

1 ~ m - k - l + l n - 1 z=l m - l + l '

where the sum is over all subsets .A _c {1, 2 . . . . . n} such that IAI = m. m--k m Finally, since lim,~oQ pro(n) = l /em!, and pro(n)" Y~4=1 (( - k - l + 1)/(m - l +

1)) < m/m! for each m > 0, we obtain

lim E(Un) = 1 + lim nE(X(1) )

m - k n E E m - k - l + l = 1 + lim - pro(n)

n~oo n 1 m -- l + l m>k /=1

e~l 1 ~m-k-£+l__ = 1 + ~" e=I m - - 7 ~ - I

by dominated convergence, and the result follows. []

The proof of Theorem 3.1 depends on a fact about the order statistics of the exponential distribution which we state as a lemma.

LEMMA 3.2. Let XI , X 2 . . . . . X m be independent exponential random variables with $

parameter 3. > O. Let S = Y~i=l X(i) be the sum of the s smallest of these m random variables. Then

( s - j - I - l ) e(s) =

j = l

PROOF. The lemma follows from the fact that the expected value o f the ith order statistic from a sample of m exponential random variables with parameter ~. is

i 1 E(X(i)) = E ~.(m - j + 1) '

j = l

and therefore

E(S) = ~.(m ---j + 1) = ~.(m - - j + 1) X ~ - m ~ - ] + l ) " i=1 '= j = I i = j j -~l

[]

4. Upper Bound. In this section we prove

THEOREM 4.1.

lim sup E(Cost(T~))

where

Fork ~ 2,

1 ~m-k-£+l _< 1 + ~ - - - - - -

m>k em! ~=1 m ---~ Sr'l l+i) + m = 0 = k - m + l I=1

k 1

k - ~ ( k _ m ) _ ( k _ l ) , = X(k) = em[

m=0

k-1 1 k ~ l )je-), ot = ot(k)= ~ J! ,

m=0 era! j=0

( ~ 1 k - ~ , ( k - m - j , ~ . J e - X ~ = ¢ ~ ( k ) = - - - ( k - 1 ) .

\m=O em[ j=0 J'( ]

The proof of the theorem is based on the analysis of a greedy algorithm. The algorithm constructs a k-map M~ for which E(Cost(M~)) can be bounded. We know that Cost(M~) < Cost(M~), so Theorems 2.1 and 2.2 together imply that E(Cost(T~)) < E(Cost(M~)) + o(1).

To construct the map Mff, we start with the optimal unrestricted random mapping, M*, and make the modifications necessary to convert it to a k-map. This is carded out in three phases as follows. In Phase 1 we start with M*, and at each vertex v with p(v, M*) > k, cut p (o, M*) - k edges into v. These edges are selected so as to minimize the "redirection cost" at the beginning of Phase 2 when the cut edges are replaced by new, more expensive edges. The redirection process may result in overfull vertices. Hence there is another round of cutting at the end of Phase 2. The edges that are cut at the end of Phase 2 get replaced by more expensive edges in Phase 3 using a simple greedy procedure.

To describe the algorithm in more detail, we need some notation. Given a map f and a vertex v, recall that 13(o, f ) = {x: f ( x ) = v}. For x ~ 13(o, f ) , let D(x, f ) be the difference between the cost of the current edge (x, f ( x ) ) and the next cheapest edge, i.e. if (x, f (x)) = ci(x), then D(x, f ) = c ( i + l ) ( X ) - c ( i ) ( x ) . If p(v, f ) = m, let DO) (v), D(2) (v) . . . . . D(m) (v) denote the order statistics of the variables {D (x, f ) : x B ( v , f ) } . In addition, i f p ( v , f ) = m > k, then let W ( v , f ) = {x: D ( x , v ) = D(j) (v, f ) for some 1 < j <_ m - k} c_ 13(v, f ) be the m - k vertices for which D(x, f ) is smallest. Finally, let jc = {v: p(v, M*) > k} and let C = {v: p(v, M*) < k}. With this notation we can describe an algorithm which constructs a heuristic mapping Mff.


Phase 1

• f = M * ; /e .(1) = o ; ~(2 ) = 0;

• For each edge (x, v) such that v ~ ~ and x ~ ~V(v, f ) , delete (x, v) from f and addx to ~ ( I ) .

Phase 2

• For all roots x ~ 7~(1), add e2(x) to f . • For every v 6 ~" that now has p(v, f ) > k, delete all edges (x, v) in f such

that x 6 R(1), and add the vertex x to 7~(2). • For every v ~ C that now has p(v, f ) > k, for every x 6 ~/V(v, f ) , delete

(x, v) from f and add x to ~(2) .

Phase 3

• For i = 1 . . . . . 1~(2)1, {Let xi be the vertex in 7~(2) with the ith smallest label. Add to f the cheapest edge from xi to a vertex that is available to xi and which has not been rejected in Phases 1-2; }

• Return f ;

I_At M~ denote the heuristic mapping that this algorithm returns. Our goal is to bound the expected cost of M~. To this end, let

n

X = E co)(v) = Cost(M*) v=l

be the cost of f at the start of Phase 1, before any edges are deleted. If el (x) is one of the edges removed in Phase 1, then el (x) will not be used in M~ and at best the edge out of x will have cost c(2)(x) in Mff. For this reason we refer to (c(2)(x) - C(l)(x)) as the "penalty for deleting et (x)." Let Y be the total penalty for deleting edges in Phase 1:

Y = E (c(2)(x) -- C(l)(X)). x~7"¢.(1)

Similarly, let W be the penalty for rejecting edges in Phase 2 and let Z be the additional penalty for adding edges in Phase 3, i.e. the cost over and above that which is accounted for by X, Y, and W. Then

(4.1) Cost(M~) = X + Y + W + Z.

PROOF OF THEOREM 4.1. Fix k > 2. From the discussion above, it is enough to bound E (Cost (M~)). We begin by noting that X + Y = U,, so from the proof of Theorem 3.1 we have

el~-~.l ~ m - k - e + l (4.2) E ( X + Y ) = 1 + ~ +o (1 ) . • e=l m - e + l

Next, to bound E(W) , we write W = ~ = l Qu, where Qu is the penalty for rejecting edges into v during Phase 2. Let p ( v ) = p (v , M*), then we have

(4.3) E (W) = ~ E ( Q o ) owl

= ~_.,{~"~Pr(p(v)=m)E(Qulp(v)=m)u=l m=o

+ Pr(p(v) > k ) E ( Q v l p ( v ) > k ) ] /

k

= n ~ Pr(p = m)E(Qu~ IP = m) + n Pr(p > k)E(Qo~ IP > k), rn=0

where vl is any fixed vertex and p = p(vD. Since Pr(p = m) --> 1~era! as n ~ c~, it is enough to determine lim supn__, ~ nE(Qo~ IP = m) and lira supn__,~ nE(Qo~ IP > k). These limits are obtained in Lemmas 4.3 and 4.4 below.

Toprovethelemmaswefirs tdefinevariables R = I'R.(1)[ and V = [{x ~ ~ ( 1 ) : e2(x) = (x, vl)}l where vl is a fixed vertex. Then we have

LEMMA 4.2. For 0 < m < k and j > O,

AJe-X lim Pr(V = j l p = m) = - -

n~oo j !

where A = A(k).

PROOF. Observe that given R = r andp = m < k, the distribution o f V isBin(r, 1 ~ ( n - 1)). So if An - k2n 3/4 < r < )~n + kZn 3/4, it follows from the Poisson approximation for the Binomial distribution that

I e-XAJ[ Cj Pr(V = j l R = r, p = m) - j----~. < - - h i ~ 4 ,

where Cj is a constant which may depend on j but which does not depend on n. Martingale concentration results (see, for example, [13] and the Appendix) establish that Pr(An - k2n 3/4 <_ R < An + k2n 3/4) > 1 - e x p ( - n 1/4) for all large n. Furthermore, since Pr(p = m) is bounded away from zero as n ---> ~ , we also have

Pr (An - k2n 3/4 < R < An + k2n3/4lp = m) > 1 - C,, exp( -n l / 4 ) ,

where Cm is a constant which may depend on m but which does not depend on n. Using these bounds, we obtain

Pr(V e-XAJ_ [ = JIP = m) - i i

< ~ Pr(V = j I R = r , p = m) {r: ]r-).nl<k2n 3/41

e-~AJ ] J! Pr(R = rip = m)

166

+ ~ Pr(V = j lR = r ,p = m) - {r : Ir-~.nl>k2n 3/4 }

<cj + 2Cm e x p ( - n 1/4)

- hi~4

and the lemma follows.

J. c . Hansen and E. Schmutz

e-X ! I - . Pr(R = rip = m)

[]

REMARK. We also note here that Pr(V = j l R = r, p = m) = (~)(1/(n - 1))J(1 -

1/(n - 1)) r--/ <__ 1~jr for all possible values o f t , so Pr(V = JtP = m) < 1/j! for all j > 0 .

LEMMA 4.3. For 0 < m < k,

oo j+m-k )Je -x j + m - k - l + l l i m n E ( Q ~ l p = m ) = y ~ J! Z - f ~ m - l _ 7 I_-1 '

n--*oo j = k - m + l / = 1

where ~. = ~.(k).

PROOF. The first step is to write

(4.4) lim nE(Qo, IP = m) tl----~ o o

n - - m

= lim ~ j = k - m + l

nE(Qo, IV = j , p = m) Pr(V = JIP = m).

We compute limn~oo nE(Q~; IV = j , p = m) by using a further conditioning argument. The calculation is divided into two cases.

C a s e l : m = O . L e t V = {x: x 6 ~ ( 1 ) andez (x ) = (x, vl)}, then the event {V = A, p = 0}, where A _ [n] and IAI = j , is measurable with respect to the a-a lgebra generated by the variables {X(i)(v): i = 1, 2, and 1 < v < n} tO {c(2)(v) - c(t)(v): 1 < v < n} (as defined in Section 1). In particular, the event {1; = A, p = 0} is independent of the variables {cO)(x) - c(2)(x): x e A}. Now for any a c [n] such that IAI = j , let Do)(A) . . . . . D(j)(A) denote the order statistics of the variables {c(3)(x) - c~z)(x) : x A}. It follows from Lemma 3.2 that

nE(Qv~I ' I )=A, I A I = j , p = O ) = nE D(i)(A) ] ) = A , I A I = j , p = O

= n E ( ~ D ( i ) ( A ) )

_ n " ~ j - k - g + l

n - 2 " - f ~ £ + I '

since the var iables {c(3)(x) - c(2)(x): x e A} are i.i .d, exp(n - 2). It fo l lows that

n (4.5) n E ( Q v , IV = j , p = 0) -

n - 2 "se~l J - k - £ + l - - - - -) - -~ ; "il < 3 j.

S i n c e P r ( V = JlP = O) < 1/j! for j > 0, the resul t now fo l lows f o r m = 0 b y L e m m a 4.2, (4.5), and domina ted convergence app l i ed to (4.4) wi th m = 0.

Case 2 : 0 < m < k. Let )2 be def ined as in Case 1 above, then the event {)) = A, (M*)-~(vl) = B , p = m}, where A, B _c [n], IAI = j , IBI ---- m, and B N A = 0, is measurab le with respec t to the t r -a lgebra gene ra t ed by the var iab les { X(i)(v): i = 1, 2, and 1 < v < n } t 0 { c ( z ) ( v ) - c ( 1 ) ( v ) : 1 < v < n , v ¢ B }. In par t icular , t h e e v e n t {)) = A, (M*)-l(vl) = B, p = m} is independen t o f t h e var iab les {c(2)(x)-c(1)(x): x B } tO {c(3) (v) - c(2) (v): v ~ A}. Let A ' = A tO B and let D(1) (A ' ) . . . . . D(m+j) (A') deno te the order stat ist ics o f the var iables {c(2) (x) - c(l) (x): x 6 B} tO {c(3) (v) - c(2)(v): v ~ A}. Then

(4.6) nE(Qo, I V = A , ( M * ) - I ( v l ) = B , I A I = j , p = m )

( J ~ k D ( i ) ( A ' ) ] ? ) = nE = A, ( M * ) - Z ( v l ) = B, [A[ = j , p = m \ i=1

= nE D(i)(A') . \ i=1

n E t % ' j + m - k D ( i ) ( A ' ) ) requi res s o m e work s ince the var iab les Now the ca lcu la t ion of ~L~i= 1 {C(2) (X) -- C(1) (X): X e B } to {CO) (V) -- C(2) (V): V ~ A} are i ndependen t but not iden t i ca l ly dis t r ibuted. In part icular , c(2)(x) - co)(x) "" exp(n - 1) for every x ~ B, whe reas c(3)(v) - c(2)(v) "~ exp(n - 2) for every v e A, so their j o in t dens i ty funct ion is g iven

by

m j+m ) f(Ul . . . . . Uj+m) = (n -- 1)m(n -- 2) j exp - - (n -- 1) Z u i -- (n -- 2) y ~ ui

i=1 i=m+l

X"~ j +m-k , on(R+)J+m.Definethefunctiong: (R+) j+m ~ R+byg(ul . . . . . Uj+m) =/-.i=l "(i) where u(i) is the i th smal les t of the coord ina tes u~ . . . . , Uj+m, Then we have

(4.7) ( ' ' )

nE ~ D(i)(A') \ i=l

= n f(R+)J+, g(u)" f (~) d~

= n f(R+):+, g(~). f (~ )d~

+ n ~R+):+= g(~)" ( f (~) - f (~)) d~,

- - x-'J+" ui) is the joint density of where f (u l . . . . . uj+m) = (n 2) j+m exp(- (n 2) z-,i=1 j + m i.i.d, exp(n - 2) random variables. By Lemma 3.2 we have

(4.8) n f g ( u ) . f ( u ) du = n J~R +)j+,, n - 2

j+m-k Z j + m - k - e + l

To bound the second term on the right-hand side of (4.7), observe that

(4.9) n ,-,--IL+)J+mg(~i)" ( f ( t ~ ) - f (~))d~

<n fR+)j+ g(~). f ( ~ ) ] l - f (~) d~ -

= n f[o,l/,/.ff]~+," g(~) . f (~) 1 - - f (~ )

+n fOoA/4_ffF+,.) g(~). f(~)[1 f ( t ~ ) d~ - f ~ •

For ~ ~ [0, 1/~cff] d+~,

I f ( ~ ) I (n- l~" ( r e l i C ( k ) (4.10) 1 - f----~ = 1 - k n _ 2 ] exp --~--'~//i ~--- ,

where C(k) is a constant which may depend on k but which does not depend on n. Thus

(4.11) n f[0, l/vrffl j+" g(u)" fi(~) ]1 -- f ~ l d~ f (~ )

C(k) f R < - - ~ - . n g(u). f (u) du +)j+m j +m -k C(k) n j + m - k - e + 1 3C(k)j

n - 2 ~-" j T m - - ~ + l qrff e=l

To bound the second term on the right-hand side of (4.9), observe that

(4.12) n f([0,1/4.gV+,) c g(~). f (~ ) 1 f (~)

j+m f(([ < 2n ~_~ ui f (~)d~ i=I O,l/~v/nlJ+m)c

C = 2n(j + m) (n - 2)u exp(- (n - 2)u) du

( 2 exp( - (n - 2)/,,/'if) ) < 4 n 2 . \ / < 8n3/2 exp(--nl/3).

It follows from (4.6)-(4.12) that

~ j + m - k - g + l

\ ,=, ,=, ¥ i

and hence

(4.13) j + m - k

n E ( Q ~ , l V = j , p = m ) - Z j + m - k - ~ + l ,=, 7 ¥ m - -

C'(k) j - ~ '

C'(k) j - ~ '

where C'(k) is a constant that may depend on k but which does not depend on j or n. The result now follows from Lemma 4.2, (4.13), and dominated convergence applied to (4.4), since Pr(V = JlP = m) < 1/j! for all j > 0. []

LEMMA 4.4.

limsupnE(Qv, lp > k)Pr (p > k) < ~.(k) • 1 - n " ' } (X)

PROOF. If p = m > k, then all edges which are mapped to ol at the start o f Phase 2 are rejected without making any comparisons. In particular, suppose that x e R(1) and e2(x) = (x, vx), then the edge e2(x) is rejected. The penalty paid for rejecting e2(x) is the minimum additional cost of finding a suitable edge out of x, i.e. the penalty for rejecting e2(x) is c(3)(x) - c(2)(x). Arguments similar to those made in the proof o f Lemma 4.3 establish that, for any A c [n],

nE(Qv, i v = a , p > k ) = n E ( ~ c ( 3 l ( x ) _ c ( 2 ) ( x ) ) nlal \xea = n - - 2 '

where ]) is as in the proof of Lemma 4.3. It follows that

(4.14) nE(Qo, Ip > k) Pr(p > k) n

= n ~ _ E ( Q v , IV = j , p > k) Pr(V = JlP > k)Pr(o > k) j=0

n£ n - 2 j P r ( V J LO > k) Pr(p > k)

j=O n

= n - 2 E(VIp > k) Pr(p > k)

= E(VIp = m) Pr(p = m), n - 2 re=k+ 1

where V = IV[. Observe that given that p = m > k, V ~ Bin(R - m + k, 1/(n - 1)) where R = I~(1)1. So

1 E(~ZIp = m) = ~ E ( R - rn + kip = m).

n - 1

170 J . C . H a n s e n and E. S e h m u t z

n Since ~7=o tdt = n = Y~t=o dr, where dt = dr(M*) denotes the number o f vertices in M* with in-degree t, we have

k - I

R = Z ( t - k)dt = Z ( k - t)dt - (k - 1)n < n. t>k t=O

T h u s

(4.15) E ( V I p = m) = 1 ~k-_.~ n - 1 (k -- t )E(d t lp = m) -- (k - 1 ) n + (m - k) t=0 n - 1

1 k - 1

< - ~ - '~ (k - t )E(dt[p = m) - ( k - 1). - n l t = 0

Now f o r 0 < t < k - 1 < m, w e m u s t h a v e E(dt lP = m) < n - 1, a n d h e n c e E ( V I p = m) < k2/2. To obtain a better bound, we note that for 0 < t < k - 1 and k < m < n - k , wehave

E(dt lP = m) = (n - 1) Pr(p(v ' ) = t ip(v1) = m) ( n - m ) ! ( 1 ) t ( 1 - 2 / n ) n-m-t

---- (n - 1)/! ( n ~ m ~ t ) ! ~1"~

e - 1 e m/n <_ ( n - l )

t! (1 - 2 /n ) k'

where v' is any vertex other than vl. Substitute this bound into (4.15) to obtain

k - I (k t ) e - l e m / n E ( V I p = m) <_ (1 - 2/n) -k Z t! - ( k - 1)

t = 0

fo rk < m < n - k. Using this bound in (4.14), we obtain

nE(Q~[p > k) Pr(p > k)

( n ( 1 _--2/_n)-k) ( ~ 1 ( k _ t ) e _ l ) < e0°g")/" t.7 - n - 2 , / \ t = o

Pr(k logn) - (k - 1)Pr(p > k)

) \ t=0 t! ( k - l ) P r ( p > k ) + o ( 1 )

=~(k) 1 - = +o(1),

k since Pr(p > logn) < e / ( logn)! and Pr(p > k) = 1 - )--~m=0 1~era! + o(1). []

It follows from Lemmas 4.3 and 4.4 that

(4.16) l i m s u p E ( W ) < m~o~m. ~ jT ] ~ m ~ - l ' - F 1 n - - * ~ = " = k - m + l " l=1

k 1

It only remains to bound E(Z) , where Z is the addit ional cost for adding edges in Phase 3, i.e. Z is the cost over and above that which is accounted for by X, Y, and W. In order to execute Phase 3, we order the elements of R ( 2 ) according to the order of their labels. Suppose that xi is the i th root in R(2 ) and let 1-' i denote the addit ional cost

incurred by adding an edge out of xi in Phase 3. So

E ( Z ) = E Fi ,

where R' = 1~(2)1. Let .4(2) denote the set of available vertices at the end o f Phase 2 and let A' = I.4(2)1, then

r

e(z) = e(r, IR' lsrsa i=1

We claim that for 1 < i < r < a,

E ( F i l R ' = r, A ' = a) <

To prove the claim, fix 1 <

= r, A' = a) Pr (R ' = r, A ' = a).

n 1 1

(n - 101ogn)(a -- i + 1) (n - 101ogn) + n 3/~'"

i < r < a. There are two cases to consider:

Case 1: X i E R ( 1 ) N ~ ( 2 ) . In this case edges el (xi) and e2(xi) have been rejected by the algorithm. In Phase 3 we examine edges e3(xi) , e4(xi) . . . . until an acceptable edge is found. Let ri denote the number of edges examined until an edge for xi is found. Note that if A~ is the number of available vertices for vertex xi , then ri corresponds to the number of draws without replacement until a black ball is drawn from an urn containing A I black balls and n - A I - 2 red balls.

Now if ri = 1, then edge e3(xi) is accepted, and in this case the addit ional cost is 0 since the incremental cost c(3)(xl) - c(2)(xi) was included in the calculation o f W. If r; = m > 1, then edge em+2(xi) is accepted and the added cost o f accepting edge em+2(Xi) is C(m+2)(Xi) -- C(3)(Xi)- So f o r n - 2 >_ m > 1, we have

E(FiIX i E 7"~(1) N 7~(2), "C i ~ - m, R ' = r, A ' = a)

= E ( C m + 2 ( X i ) - - C3(Xi)[Xi E ~ ( 1 ) N ~ ( 2 ) , ~i = m , e ' : r, A ' = a)

1 1 1

n _ 3 + ~-S-~_4 + " " + n - l - r n m - 1

< + 2 log n • l[m>61ogn]. - n - 1 0 1 o g n

I t follows from the proof of Lemma 2.9, that IA~I > n / 4 , so Pr(r i > 61ogn) < (3h61ogn 1/n3/2 a , < and thus

E(Fi IXi E ~r~(1) ~ 7~(2), e ' = r, A ' = a )

E(r i l x i ~ R(1) N 7~(2), R ' = r, A ' = a) 1 21ogn < + ~ - n - 101ogn (n -- 101ogn) n3/2 "

Now given A~ = ai, standard results fo r sampling without replacement yield

n n E(rilxi ~ R(1) M R(2 ) , R ' = r, A' = a , A I = ai) < - - <

ai a - i + l

always. The last inequality follows from a s imple observation: i f there are a available vertices at the start of Phase 3, then as each root finds a vertex the number of available vertices is reduced by at most 1, so A' i > A' - i + 1 always. It fol lows that

n E(rilxi ~ ~ ( 1 ) t3 ~ ( 2 ) , R ' = r, A' = a) <

a - i + l

and hence

E(Filxi ~ 7E(1) N 7"¢(2), R' = r, A' = a)

n 1 2 log n

< (n - 101ogn)(a - i + I) n -- 101ogn + n 3/-'-----~"

Case 2: xi ~ R(2) \7~(1) . In this case edge el(xi) has been rejected by the a lgor i thm during Phase 2. In Phase 3 edges eE(xi), e3(xi) . . . . are examined until an acceptable edge is found. As in Case 1, Fi = 0 if ri = 1, and for n - 1 _> m > 1,

E(Filxi ~ 7~(2)\7~(1), ri = m, R ' = r, A' = a)

-~- g ( C m + l ( X i ) - - C2(Xi)lXi E ~ ( 2 ) \ 7 ~ ( 1 ) , ri = m, R' = r, A' ---- a ) 1 1 1

- - n - 2 + ~ - 3 + ' ' ' + - n - m m - 1

< + 2 log n • l I k > 6 1 o g n ] . - n - - 1 0 1 o g n

T h e same argument as given in Case 1 yields

E(Fi lx i E ~(2) \T~(1) , R' = r, A' = a) n 1 21ogn < - b ~

- (n - 101ogn)(a - i + 1) n - 101ogn n 3/2

So in all cases, for i < r , we have

n 1 2 log n E(Fi[R ' = r, A' = a) <_ + - -

(n - 101ogn) (a - i + 1) n - 101ogn n 3/2 "

Thus

~ - - - ~ £ ---- A' = = A ' E (Z) < E(Fi IR ' r, a) Pr (R ' r, = a ) l<_r<_a i=1

< )---, ~ ( n 1

l_<~_<a i=l (n -- l O l o g n ) ( a - - i + 1)

x Pr (R ' ---- r, A' = a)

n A ~ < - - n _ l O l o g n ( E ( I ° g ( A , - - _ - ~ ) ) E( -Rn ' ) )+

2 1 o g n ~

n - 101ogn + n- - -~- - ' ]

2 E ( R ' ) log n n3/2

In the Appendix we show that there is a constant Ck, which may depend on k but which does not depend on n, such that for the event

y = {IA' - not(k)l _< Ckn 3/4, IR' -- nfl(k)l <_ Ck n3/4}

we have

Pr(y) > 1 - 4 e x p ( - n l / 4 ) .

It follows that for all large n,

(4.17) E(Z) <_ l o g \ o t ( k ) _ f l ( k ) ] - / ~ ( k ) + o ( 1 ) Pr(y)+21og(n)Pr(]/c)

= log ff(__k) o(1) _

and Theorem 4.1 now follows from (4.2), (4.16), and (4.17). []

It is not difficult to describe various ways to improve the algorithm for constructing M~. However, with each improvement of the algorithm, the analysis of the algorithm becomes more complicated.

5. Conclusions. Table 1 summarizes our results for a few small values o f k. Limited simulation data for 200 to 400 vertices suggest that, in the case k = 2, the lower bound is sharper than the upper bound. In this paper we have considered exp(1) costs on the edges. It likely that, as in the k = oo case [8], the arguments can be carried out for other distributions as well, provided the order statistics of the chosen edge distribution are well behaved.

We note that for exp(1) edge costs, Aldous [14] has shown that the expected cost of the optimal assignment converges to a constant as n --~ oo. This result also holds in the case k = oo for fairly general edge distributions. We speculate that a similar result might hold 2 < k < oo. The methods used in the case k = oo are insufficient to prove this, but it may be possible to adapt Aldous' "objective method" to obtain the result. In the case k = c~, the limiting constant can be determined, but it is not known for the assignment problem with either exp(1) or U(0, 1) edge costs.

Table 1. Summary of results.

Asymptotic bounds

k Lower Upper

2 1.03892136 1.06806181 4 1.00097152 1.00100255 5 1.00012721 1.00012800 6 1.00001487 1.00001490 7 1.00000156 1.00000157

Acknowledgments. Our first version of this paper used uniform U(O, 1) edge costs. We thank the anonymous referees for suggesting exp(1), and also for catching an error in one of our proofs. The authors also thank Ljubomir Perkovic for use of his computer and optimization software.

Appendix. Let Ilk be the following computational problem: given as input a cost matrix C, find a minimum cost k-tree. Then we have

LEMMA A. 1. Ilk is NP-hard.

PROOF. Let Uk be the restriction of Ilk to symmetric matrices whose entries are all zeros and ones. The instances of Uk correspond, in an obvious way, to instances of the NP-complete degree-k constrained spanning tree problem for undirected graphs [15, page 206, comment to ND1]. The correspondence is: include an edge {i, j} in the undirected graph G if and only if Cost((i, j ) ) = Cost((j, i)) = 0. Then G has an undirected degree-k spanning tree iff the optimal (directed) k-tree for C has cost 0. Thus the existence of a polynomial time algorithm for Ilk would imply the existence of a polynomial time algorithm for an NP complete problem. []

In the remainder of this appendix, we prove the martingale concentration results used in the proofs of the main theorems.

LEMMA A.2. L e t M * b e a u n i f o r m r a n d o m m a p p i n g o n { l , 2 , . . . , n } , t h e n f o r O < k < n and t > O,

Pr(ldk - E(dk)l > t) < 2exp - ,

where dk = dk(M*) denotes the number of vertices with in-degree k in M*.

PROOE The proof of the lemma is an application of Azuma's inequality for martingale differences, and is based on the ideas of Steele [ 13]. Let ZI, Z2 . . . . . Zn be i.i.d, random variables such that Zi corresponds to the "choice" made by vertex i, i.e. Zi = j if and only if M*(i) = j . We Write d, = dk(Zl, Z2 . . . . . Zn) to emphasize the dependence of dk on the variables Zl, Z2 . . . . . Zn which determine the mapping M*. Next, let )r. = a(Zl . . . . . Zi) denote the a-algebra generated by the variables Z1, Z2 . . . . . Zi and let )r0 denote the trivial tr-algebra. The variables E (d~ I.T'0), E (dk ].T'I) . . . . . E (dk I.T'~) form a martingale with E(dklJr0) = E(dk) and E(dklU,) = dk, and the variables Wi = E(dkl~.) - E(dk[~'/_l), i = 0, 1, 2 . . . . . n, form a martingale difference sequence. Since dk -- E(dk) = ~]i~1 Wi, Azuma's inequality for martingale differences tells us that

P r ( I d k - E ( d k ) [ > t ) = P r ~ W i > t < 2 e x p - ,_- i - 2 W w ,

So the lemma follows provided that II Wi 112 < 1 for all 1 < i < n.

To bound on [I Wi IIZ~ we use the following trick. Let Zs, 2~2 . . . . . Zn be a sequence of i.i.d, random variables which are uniform on {1, 2 . . . . . n} and which are independent of the variables Z5, Z2 . . . . . Zn. We claim that

E(dk(Z5 . . . . . Z i . . . . , Zn)}~ ' -5) = E(dk(Z5 . . . . . Z i - - 5 , Z i , Z i + l . . . . . Zn) l~ / ) .

To see this, note that the conditional distribution of dk (Z1 . . . . . Zi_ 5, Zi, Zi+l, . . . , Zn) given ~'i is the same as the conditional distribution ofd t (Z5 . . . . . Z i - 5, Zi, Zi+j, . . . , Z , ) given .Tq,._l since the ~r-algebra ~/ is independent of Zi and only gives us information about Zt , Z2 . . . . . Z i - l . Thus we can write

Wi = E(d~(Z5 . . . . . Zi . . . . . Z . ) - dk(Z5 . . . . . Z i -5 , Zi , Zi+5 . . . . . Z . ) l ~ / ) ,

i.e. Wi can be written as a single conditional expectation with respect to ~ / . Now it is easy to see that (without conditioning)

Irk(z1 . . . . . Zi . . . . . Z , ) - ak(z1 . . . . . Z i - l , Zi , Zi+5, . . . , Z,) I _< 1

since all "choices" except that made by vertex i, are the same and the choices made by Zi and Zi can only create a discrepency of at most 1 in the count of the number of vertices with in-degree k. Conditioning does not increase the L ~ bound, so we get II Wa I1~ < 1 too. It follows from Azuma 's inequality that for any ~. > 0,

([~"]~ I ) ( - t 2 ) Pr ( Idk( f ) - E ( d k ( f ) ) l > t) = Pr W i > t < 2exp - ~ n " []

i=5

As an application of Lemma A.2 we have

COROLLARY A.3. For all large n and f ixed k > 2,

( __n[ ) Pr d m - em! < n3/4' 0 < m < k > 1 - e x p ( - n 1/4)

and

Pr(IR - n~.(k)l < k2n 3/4) > 1 - e x p ( - n t / 4 ) ,

where R I~(1)[ and X(k) k-5 = = Y'~m=0(k - m ) / e m ! - (k - 1).

PROOF. Since E(dm) = n(~,)(1/n)m(1 - l / n ) n-m for 0 ~ m < k, there is a constant Ck which does not depend on n such that IE(dm) - n/ern!l < Ck for 0 < m < k. Applying L e m m a A.2, we obtain

- - > n 3 /4 ~ Pr [dra - E(dm)[ > < 2exp

for 0 < m < k and this establishes the first part of the result. Now recall that

k e = 1~(1)1 = Z ( m - k)dm = Z ( k - m)dm - - (k - 1)n

m > k m=0

n since )--'.n=l mdm = n = ~_,m=odm, so the event {Idm - n/em!l < n 3/4, 0 < m < k} is contained in the event {I R - n~.(k)[ < k2n3/4}, and the result follows. []

Next, recall that .4(2) denotes the set o f available vertices at the start o f Phase 3 and 7~(2) denotes the roots at the start of Phase 3 in the algorithm from Section 4. Then adapting the proof of Lemma A.2, we obtain

LEMMA A.4. For all large n and fixed k > 2, there exists a constant Ck which does not depend on n, such that

Pr(IA' - ot(k)n[ < Ckn 3/4, JR' - ~(k)nl < Ck n3/4)

> 1 - 4 e x p ( - n l / 4 ) ,

where A' = IA(2)I, R' = I~(2)1,

k - I l k - m - I ~ j e _ X

= j o J ! ' m=O " "=

/~(k) = (~ -~ 1 k ~ l (k - - m -- J)3Je-~ ~ -- (k --1) , \m=0 em"-~, j=0 7 ! ]

and )~ = ~.(k) (as defined in Corollary A.3).

PROOF. We define the random variable

k - I ~ - m - , ( R ) ( 1 ) J ( n _ 2 ~ R - j g ( d o , d , . . . . . d k - l ) = ; '

m = 0 j = 0

where R = [~(1)l is the number of roots at the end of Phase 1. Standard calculations and approximations establish that if Id,, - n /em !1 < n3/4 for 0 <_ m < k, then {R - n~.(k){ < kZn 3/4 and there is some constant C~ which does not depend on n such that

[g(~r) _ not(k)l < Ctk n3/4 .

We show that

(A.1) P r ( ] A ' - g ( d ) [ < n 3/4, dm e~ . ] ) - - < n 3 / 4 , 0 < m < k > 1 - - 2 e x p ( - - n 1/4)

and, since

- < 0 < m < k c_ { IA ' -na(k ) l <_ (C'k+l)n3/4},

we obtain

(A.2) Pr ( [A ' -no~(k ) l < (C~-t-1)n 3/4) > 1 - 2 e x p ( - n l / 4 ) .

Let .,4(1) denote the set of available vertices at the end of Phase 1 and let ~ denote the event that .,4(1) = A, ~ ( 1 ) = B, and M* = f , where f is a mapping such that

Idm(f) - n/em! l < n 3/4 for 0 < m < k and A and B are subsets of vertices such that IBI < IAI. We claim that E(A' I~) = g ( d ( f ) ) . To see this, suppose that v e A and p(v , f ) = m. Then v e ,,4(2) only if the number of roots in B which are mapped to v in Phase 2 is less than k - m, so

k-m-l(;)( 1 )J(n--2~ r-j Pr(v ~ A(2)[~, p(v , f ) = m) = j~=o ~ k,n - 1 ) '

where r = IBi = 17~(1)1. It follows that

E(A' I~) -- y ~ P r ( v e A(2)I~) yEA k-1 k-m-l(~) ( 1 ) j (n_2~r - j

= E dm(f) E ~ ~k?l_ 1J ,n=0 j=0

= g ( d ( f ) ) ,

(Note that given the event ~, the variable g ( d ( f ) ) = g (do ( f ) , d l ( f ) . . . . . & ( f ) ) is completely determined, since we have conditioned on M* = f . )

Now fix ~ and let P~ denote the conditional probability measure P (. I~). Let xl , x2 . . . . . xr denote the vertices in B and let Zi denote the vertex that xi is mapped to in Phase 2, i.e. Z i = 1) i fez(xi) = (xi, v). Now given the event ~, the variables Z t , Z2 . . . . , Z, are independent and each variable Zi is uniformly distributed over the set { 1, 2 . . . . . . n}k{ f (x i ) } . In this (conditional) probability space, let .T~ = tr (ZI , Z2 . . . . , Zi) denote the (r-algebra generated by the variables Z1, Z2 . . . . . Zi and let .To denote the trivial o-algebra. Let Wk = E~ (A ' I~ / ) - E¢ ( A ' I ~ ' - I ) . So Azuma 's inequality tells us that

( n',2 P ~ ( I A ' - g ( d ( f ) ) l > n3/4) = P ~ ( I A ' - E~(A')I > n 3/4) _< 2exp 2 ~ W i l I 2 ] .

As in the proof of Lemma A.2, let 2~1, Z2 . . . . . Zr be a sequence of independent random variables such that Zi "~ Zi for 1 < i < r and which are also independent of the variables Zl , Z2 . . . . . Z~. Then

Wi = E~(A'(ZI . . . . . Zi . . . . . Zr) -- A ' (ZI , . . . , Zi-1, Z i , Z i + l . . . . . Z r ) l . T i )

and [J Wi [Ioo _< 1 for 1 n 3/4) __< 2exp ~k----~y ) _~< exp(- -n '/4)

since r = IBI < n. It follows from this and Lemma A.2, that

Pr ( I n ' - g(g')l < n 3/4, dra - - - n I < n3/4, 0 < m < k ) em! - -

= ~ P r ( I A ' - g ( d ( f ) ) l < n3/41~)Pr(~)

___ Z ( 1 - exp( -n l /4 ) ) Pr(~)

= ( 1 - e x p ( - n l / 4 ) ) P r ( d,n -- ~ < n3/4, 0 <m<k_ )

> (1 -- exp(-n l /4) ) (1 - exp(--nl/4)),

where the summation is over all events ~ such that Idm - n/em!l < n 3/4, 0 _< m ~ k. This establishes inequality (A. 1) and hence (A.2).

Similar calculations yield a concentration result for R'. In this case we define the function

~(do, dl . . . . . dk)

~ [ [k-m'l ( R ) ( 1 ) J { n - 2 ) R - j + ( k - m ) )

= dm (k -m- i ) n-l: - (k - i ) . m =0 j =0

k - I (Note: we have used the identity R = )'-~.m=0(k - m)dm - (k - 1).) Again, rou- tine approximations establish that if [din - n/em![ < n 3/4 for 0 < m < k, then Ig(d0 . . . . . dk) -- n~(k)l < C~kn 3/4 for some constant C~ which does not depend on n. So, as in the derivation of inequality (A.2), it is enough to show that

Pr IR' - g(d)l < n3/4, - < 0 < m < k > 1 - 2 e x p ( - n l / 4 ) .

Again, let ~ denote the event .A(1) = A, ~ ( 1 ) = B, and M* = f , where f is a mapping such that ]din(f) - n/em!l < n 3/4 for 0 < m < k, and A and B are subsets o f vertices with IBI < IAI. We claim that

E~(R') : = E(R' [~) = ~ ( d ( f ) ) .

To see this, note that although it is not necessarily the case that ~ ( 2 ) ___ R(1) , it is always the case that R' < R. In particular, the overall number of roots is only reduced by mapping vertices in B to available vertices in .A(1) = A during Phase 2. Redirecting a vertex x 6 B to an unavailable vertex during Phase 2 may result in deleting x from the set of roots, but in this case another vertex is added to the set of roots and the number of roots is not decreased. For each v 6 A, let Yv denote the number of roots which are subtracted from 17~(1)1 = IBI = r due to the roots in B which are mapped to v in Phase 2, then given the configuration ~, we can write

R' = r - Z Yv. yEA

I f p ( v , f ) = m < k, then Y~ i s a t m o s t k - m a n d f o r 0 < j _< k - m - 1,

P r ( Y o = j [ ~ , p ( v , f ) = m ) = ~ \ n - l ] '

whereas

Pr(Y~ = k - m}~, p(v , f ) = m) = 1 -

It follows that

k - m - I

( ; ) ( n - - ~ ) j,n-(n-2"~r-j'lj

, - m - I ( ; ) ( 1 ) j ( n _ 2 , ~ r - j E(YoI~, p(v , f ) = m) = j~=o ( j - k + m) ~ \ n - 1 ,] + (k - m)

and hence

/ P ~ ( I R ' - g ( d ( f ) ) l > n3/4) = P ~ ( I R ' - E ~ ( R ' ) I > n 3/4) _< 2exp

It is straightforward to check that ItW~ll~ _< 1 for 1 < i < r, so

since E(RI~) = r = z..,,n=O~V'~-I tk - m ) d m ( f ) - (k - 1). Now let xl, x2, .. • ,xr denote the vertices in B. For 1 < i < r, let Zi denote the vertex that xi is mapped to in Phase 2. Let ~i = tr (Z1, Z2 . . . . . Zi) denote the a-algebra generated by the variables Zl , Z2 . . . . . Zi and let 5% denote the trivial er-algebra. Let W[ = E~(R'I~i) - E ¢ ( R ' I ~ . - D , then Azuma's inequality tells us that

n 3/2 "~ r--'i~Wt 2 " 2 Z i = I 1] i1100 /

Thus

n3/2,) < P~(iR' - g(t~(f)) I > n 3/4) < 2exp ~ , - -~"r J - exp(-n] /4)"

( °l ) Pr IR' - ~(d)t < n 3/4, em! < n3/4' O < m < k

> ~--~(1 - e x p ( - n l / 4 ) ) P ( ~ )

( 1 - exp ( -n l /4 ) ) pr (I dm - e ~ . ) = < n 3/4, 0 < m < k

> (1 - exp(--nl/4))(1 -- exp(--n]/4)),

where the summation is over all events ~ such that Idm(f) - n/em!l < n 3/n for 0 < m < k. Finally, since

[ dm --n I ] IR' - g ( d ( f ) ) l < n3/4' - em! < n3/4' O < m < k

_c {IR' - n/~(k)l < (C~ + 1)n3/4},

we obtain

(A.3) Pr(IR' - n/3(k)[ < (C~ + 1)n V4) > 1 - 2 e x p ( - n ] / 4 ) .

The lemma follows from inequalities (A.2) and (A.3). [ ]

180 J. C. Hansen and E. Schmutz

Note Added in Proof. W e r e c e n t l y l e a r n e d o f D o t s e n k o ' s r e s u l t s [ 1 6 ] f o r t h e c a s e k = 1.

References

[1] M.X. Goemans and M.S. Kodialam, A lower bound on the expected cost of an optimal assignment, Math. Oper. Res. 18 (1993), 267-274.

[2] B. Olin, Asymptotic properties of random assignment problems, Ph.D. thesis, Kungl Tekniska Hogskolan, Stockholm, 1992.

[3] D. Coppersmith and G. Sorkin, Constructive bounds and exact expectations for the random assignment problem, Random Struct. Algorithms 15(2) (1999), 113-144.

[4] A.M. Frieze• On the va•ue •f a rand•m minimum spanning tree pr•b•em• Discrete Appl. Math. • • ( • 985)• 47-56.

[5] A. Beveridge, A.M. Frieze, and C.J.H. McDiarmid, Random minimum length spanning trees in regular graphs, Preprint.

[6] A.M. Frieze and C.J.H. McDiarmid, On random minimum length spanning trees, Combinatorica 9(4) (1989), 363-374.

[7] S. Khuller, B. Raghavachari, and N. Young, Low-degree spanning trees of small weight, SlAM J. Comput. 25 (1996), 355-368.

[8] J.C. Hansen, Limit laws for the optimal directed tree with random costs, Combin. Probab. Comput. 6 (1997), 315-335.

[9] C. McDiarmid, On the greedy algorithm with random costs, Math. Programming 36 (1986), 245-255. [ 10] M.S. Bazaraa, J.J. Jarvis, and H.D. Sherali, Linear Programming and Network Flows, 2nd edn., Wiley,

London, 1990, page 481. [l 1] R.M. Karp and M. Steele, Probabilistic analysis of heuristics, in The Traveling Salesman Problem, ed.

Eugene L. Lawler et al., Wiley, London, 1985, pages 181-205. [12] P. Flajolet and M. Soria, General combinatorial schemas with Gausssian limit distributions and expo-

nential tails, Discrete Math. 114 (1993), 159-180. [13] J.M. Steele, Probability in Combinatorial Optimization, SIAM, Philadelphia, PA, 1997. [ 14] D. Aldous, Asymptotics in the random assignment problem, Probab. Theory Appl. 93 (1992), 507-534. [15] M.R. Garey and D.S. Johnson, Computers and Intractability, Freeman, San Francisco, CA, 1979,

page 206. [ 16] V.S. Dotsenko, Exact solution of the random bipartite matching model, J. Phys. A 33 (2000), 2015-2030.

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