Queues & Priority QueuesQueues & Priority Queues

Radix SortRadix SortHeap SortHeap Sort

OutlineOutline

QueuesQueues• queue operationsqueue operations• algorithms with queuesalgorithms with queues• radix sortradix sort

Priority QueuesPriority Queues• priority queue operationspriority queue operations• the “heap” data structurethe “heap” data structure• heap sortheap sort

Queue OperationsQueue Operations

Queues require these operationsQueues require these operations• add at back (add at back (enqueueenqueue, , putput, , insertinsert)) .add(item).add(item)• delete from front (delete from front (dequeuedequeue, , getget)) .remove().remove()

Typically have some othersTypically have some others• check if emptycheck if empty .isEmpty().isEmpty()• inspect front element (inspect front element (getFrontgetFront, , peekpeek)) .front().front()• empty it outempty it out .clear().clear()• get sizeget size .size().size()

A QueueA Queue

Arranged left to rightArranged left to right• front at the leftfront at the left

Old items are Old items are removedremoved or or dequeued dequeued at frontat front• myQueue.remove()myQueue.remove()

New items are New items are inserted inserted or or enqueued enqueued at backat back• myQueue.add(99)myQueue.add(99)

17 24 81 12 5 7 17 12 8

Queue

Front

Back

A QueueA Queue

Arranged left to rightArranged left to right• front at the leftfront at the left

Old items are Old items are removedremoved or or dequeued dequeued at frontat front• myQueue.remove()myQueue.remove()

New items are New items are inserted inserted or or enqueued enqueued at backat back• myQueue.add(99)myQueue.add(99)

24 81 12 5 7 17 12 8

Queue

Front

Back

A QueueA Queue

Arranged left to rightArranged left to right• front at the leftfront at the left

Old items are Old items are removedremoved or or dequeued dequeued at frontat front• myQueue.remove()myQueue.remove()

New items are New items are inserted inserted or or enqueued enqueued at backat back• myQueue.add(99)myQueue.add(99)

24 81 12 5 7 17 12 8 99

Queue

Front

Back

Real World QueuesReal World Queues

Any line-upAny line-up• bank, ticket window, amusement ridebank, ticket window, amusement ride• (except – people might give up & leave)(except – people might give up & leave)

Assembly lineAssembly line• chassis inserted at one endchassis inserted at one end• completed vehicle comes off the othercompleted vehicle comes off the other

FILO and LILOFILO and LILO

Stack operations all at one endStack operations all at one end Last element in the Last element in the stackstack is the first one out is the first one out

• Last In, First Out = LIFOLast In, First Out = LIFO• AKA FILO = First In, Last OutAKA FILO = First In, Last Out

Last element in a Last element in a queuequeue is last one out is last one out• Last In, Last Out = LILOLast In, Last Out = LILO• AKA FIFO = First In, First OutAKA FIFO = First In, First Out

ExerciseExercise

Draw the Draw the queuesqueues that result from the that result from the following operationsfollowing operations

Start with empty each timeStart with empty each time• add A, add B, add C, remove, removeadd A, add B, add C, remove, remove• add 1, add 2, remove, add 3, add 4add 1, add 2, remove, add 3, add 4

Exceptional CircumstancesExceptional Circumstances

Similar to stacks:Similar to stacks:• try to remove/get front of empty queuetry to remove/get front of empty queue• try to add to a full queuetry to add to a full queue• try to add a nulltry to add a null

Solutions similar as well:Solutions similar as well:• return null / throw an exceptionreturn null / throw an exception• return false / throw an exceptionreturn false / throw an exception• two versions of remove and fronttwo versions of remove and front

Using QueuesUsing Queues

File serverFile server Railway sidingsRailway sidings Wire routing (shortest path)Wire routing (shortest path) Image component labelingImage component labeling Radix SortRadix Sort

Wire Routing ProblemWire Routing Problem

Need to connect two Need to connect two locations on a devicelocations on a device• can’t go over existing can’t go over existing

componentscomponents• want shortest pathwant shortest path

00000065456001110654311001110543S1F0000011143450001111051110001110001110000000011110000000000000000011110000000011110000000000000000

Wire Routing ProblemWire Routing Problem

Represent device by 0-1 Represent device by 0-1 matrixmatrix• 0 = free space0 = free space• 1 = component1 = component

Need start & end pointsNeed start & end points• locations in matrixlocations in matrix• (2, 8) and (2, 10)(2, 8) and (2, 10)

000000000000000000000000011100000110011100000110011100000100011100000100000011100000000011100000001111001110001111001110001110001110001110001110000000011110000000011110000000000000000000000000000011110000000011110000000011110000000011110000000000000000000000000000

Wire Routing ProblemWire Routing Problem

Two Pass MethodTwo Pass Methodto RouteWire(matrix, start, end)to RouteWire(matrix, start, end)

// find distances from start// find distances from startLabelDistances(matrix, start, end)LabelDistances(matrix, start, end)

// find path from end back to start// find path from end back to startFindRouteBack(matrix, start, end)FindRouteBack(matrix, start, end)

Distance PassDistance Pass

add start cell, labeled 2add start cell, labeled 2• [(2, 8)][(2, 8)]

remove start cell (=2), add remove start cell (=2), add its neighbours, labeled 3its neighbours, labeled 3• [(1, 8), (3, 8), (2, 7)][(1, 8), (3, 8), (2, 7)]

remove next cell (=3), add remove next cell (=3), add itsits neighbours, labeled 4 neighbours, labeled 4• [(3, 8), (2, 7), (0, 8), (1, 7)][(3, 8), (2, 7), (0, 8), (1, 7)]

0000000000000000000000000011111100000000001111000011111100000000ss11ee00000000001111110000000000000011111111000011111100000011111100000011111100000000000000001111111100000000000000000000000000000000001111111100000000000000001111111100000000000000000000000000000000

00000000000000000000000000111111000000000011110000111111000000002211ee00000000001111110000000000000011111111000011111100000011111100000011111100000000000000001111111100000000000000000000000000000000001111111100000000000000001111111100000000000000000000000000000000

0123456789ab0123456789a

00000000000000000000000000111111000000003311110000111111000000332211ee00000000001111110033000000000011111111000011111100000011111100000011111100000000000000001111111100000000000000000000000000000000001111111100000000000000001111111100000000000000000000000000000000

00000000000000004400000000111111000000443311110000111111000000323211ee00000000001111110033000000000011111111000011111100000011111100000011111100000000000000001111111100000000000000000000000000000000001111111100000000000000001111111100000000000000000000000000000000

Distance PassDistance Pass

Repeat until you reach goalRepeat until you reach goal• or until queue is emptyor until queue is empty

Remove (3, 10), labeled 5, Remove (3, 10), labeled 5, label its neighbours 6label its neighbours 6• [(4, 7), (2, 5), (0, 10), …][(4, 7), (2, 5), (0, 10), …]• add (2, 10) = endadd (2, 10) = end• can stop nowcan stop now

000000000000654566545600001111110065436543111100001111110054325432116600000000001111114344345500000011111111005511111100000011111100000011111100000000000000001111111100000000000000000000000000000000001111111100000000000000001111111100000000000000000000000000000000

0123456789ab0123456789a

Label Distances CodeLabel Distances Code

to LabelDistances(matrix, start, end)to LabelDistances(matrix, start, end)matrix[start] matrix[start] 2; 2;q.add(start);q.add(start);while (!q.empty())while (!q.empty())

if (q.front() = end)if (q.front() = end) break;break;LabelNeighbours(q.top(), matrix);LabelNeighbours(q.top(), matrix);q.remove();q.remove();

if (q.empty())if (q.empty())throw exception(“No path found”);throw exception(“No path found”);

Doesn’t stop as early as it could

Path FindingPath Finding

Start at goalStart at goal Pick a neighbour labeled Pick a neighbour labeled

one less than youone less than you Repeat until reach startRepeat until reach start

• in this example a single path in this example a single path

000000000000654566545600001111110065436543111100001111110054325432116600000000001111114345434500000011111111005511111100000011111100000011111100000000000000001111111100000000000000000000000000000000001111111100000000000000001111111100000000000000000000000000000000

Finding Path BackFinding Path Back

to FindRouteBack(matrix, start, end)to FindRouteBack(matrix, start, end)current current end; end;while (current != start)while (current != start)

label label matrix[current]; matrix[current];matrix[current] matrix[current] 1; 1; // now used// now usedcurrent current FindNeighbour(matrix, current, FindNeighbour(matrix, current,

label–1);label–1);matrix[start] matrix[start] 1; 1; // start used, too// start used, too

ExercisesExercises

Show how the queue develops for the rest Show how the queue develops for the rest of the example aboveof the example above• assume always look north, east, south, westassume always look north, east, south, west• assume stop as soon as the end cell labeledassume stop as soon as the end cell labeled

To think aboutTo think about• how could we rewrite the code above to stop as how could we rewrite the code above to stop as

soon as the end cell gets labeled?soon as the end cell gets labeled?

Radix SortRadix Sort

An array of numbers to sortAn array of numbers to sort• [75, 73, 72, 64, 115, 31, 232, 105][75, 73, 72, 64, 115, 31, 232, 105]

A List of 10 queuesA List of 10 queuesList<Queue<Integer>> radix = …;List<Queue<Integer>> radix = …;

0 1 2 3 4 5 6 7 8 9

Radix SortRadix Sort

Use 1s digit to choose queue to put them inUse 1s digit to choose queue to put them in• [7[755, 7, 733, 7, 722, 6, 644, 11, 1155, 3, 311, 23, 2322, 10, 1055]]

A List of 10 queuesA List of 10 queuesList<Queue<Integer>> radix = …;List<Queue<Integer>> radix = …;

3123272 73 64

10511575

0 1 2 3 4 5 6 7 8 9

Radix SortRadix Sort

Recombine in order from 0 to 9Recombine in order from 0 to 9• [3[311, 7, 722, 23, 2322, 7, 733, 6, 644, 7, 755, 11, 1155, 10, 1055]]

A List of 10 queuesA List of 10 queuesList<Queue<Integer>> radix = …;List<Queue<Integer>> radix = …;

3123272 73 64

10511575

0 1 2 3 4 5 6 7 8 9

Radix SortRadix Sort

Repeat using 10s digitRepeat using 10s digit• [[331, 1, 772, 22, 2332, 2, 773, 3, 664, 4, 775, 15, 1115, 15, 1005]5]

A List of 10 queuesA List of 10 queuesList<Queue<Integer>> radix = …;List<Queue<Integer>> radix = …;

105 115 23231 64

757372

0 1 2 3 4 5 6 7 8 9

Radix SortRadix Sort

Recombine in order from 0 to 9Recombine in order from 0 to 9• [1[1005, 15, 1115, 5, 331, 21, 2332, 2, 664, 4, 772, 2, 773, 3, 775]5]

A List of 10 queuesA List of 10 queuesList<Queue<Integer>> radix = …;List<Queue<Integer>> radix = …;

0 1 2 3 4 5 6 7 8 9

105 115 23231 64

757372

Radix SortRadix Sort

Repeat using 100s digitRepeat using 100s digit• [[1105, 05, 1115, 31, 15, 31, 2232, 64, 72, 73, 75]32, 64, 72, 73, 75]

A List of 10 queuesA List of 10 queuesList<Queue<Integer>> radix = …;List<Queue<Integer>> radix = …;

7573726431

115105 232

0 1 2 3 4 5 6 7 8 9

Radix SortRadix Sort

Recombine in order from 0 to 9Recombine in order from 0 to 9• [31, 64, 72, 73, 75, 105, 115, 232][31, 64, 72, 73, 75, 105, 115, 232]

A List of 10 queuesA List of 10 queuesList<Queue<Integer>> radix = …;List<Queue<Integer>> radix = …;

7573726431

115105 232

0 1 2 3 4 5 6 7 8 9

Radix SortRadix Sort

Quit when no number has the digit you’re Quit when no number has the digit you’re looking forlooking for• find maximum & go until it has been passedfind maximum & go until it has been passedfor (r = 1; max <= r; r *= 10) {for (r = 1; max <= r; r *= 10) {

splitList(arr, radix, r);splitList(arr, radix, r); // arr // arr radix radixrecombine(arr, radix);recombine(arr, radix); // radix // radix arr arr

}}

Complexity of Radix SortComplexity of Radix Sort

Number of operations:Number of operations:• split queue: linear in list lengthsplit queue: linear in list length• recombine queues: linear in list lengthrecombine queues: linear in list length• how many times do we split/recombine?how many times do we split/recombine?

» doesn’t depend on length of the doesn’t depend on length of the listlist

Linear in length of listLinear in length of list• looks better than quadratic or N looks better than quadratic or N loglog N N

» but not necessarily better….but not necessarily better….

Problems for Radix SortProblems for Radix Sort

Depends on “length” of biggest itemDepends on “length” of biggest item• 6-digit numbers take twice as long to sort as 3-6-digit numbers take twice as long to sort as 3-

digit numbersdigit numbers• 80-character strings take ten times as long to 80-character strings take ten times as long to

sort as 8-character stringssort as 8-character strings» and need 94 or more queuesand need 94 or more queues

Doesn’t work on numbers with decimalsDoesn’t work on numbers with decimals» unless write them into long Strings and then read unless write them into long Strings and then read

them back again afterwardsthem back again afterwards

Priority Queue OperationsPriority Queue Operations

Priority queues must have these operations:Priority queues must have these operations:• add (add (pushpush, , insertinsert)) .add(item).add(item)• remove highest priority item (remove highest priority item (poppop)) .remove().remove()

Typically have more operations….Typically have more operations….• check if emptycheck if empty .isEmpty().isEmpty()• look at highest priority itemlook at highest priority item .peek().peek()• get sizeget size .size().size()• make emptymake empty .clear().clear()

Priority QueuesPriority Queues

Priority queue is a queue where the “most Priority queue is a queue where the “most important” items get removed firstimportant” items get removed first• ““most important” = “highest priority”most important” = “highest priority”

Most important is usually the smallestMost important is usually the smallest• priority #1priority #1• priority #2priority #2• ……• priority # 1,000,004priority # 1,000,004

Priority QueuesPriority Queues

The The smallestsmallest element is removed element is removed

• pq.remove();pq.remove();

• pq.remove();pq.remove();

• pq.remove();pq.remove();

5 7 8 12 17 12 17 24 81

7 12 8 24 17 12 17 81

8 12 12 24 17 81 17

12 17 12 24 17 81

Priority QueuesPriority Queues

Additions made anywhere convenientAdditions made anywhere convenient

• pq.add(75); pq.add(75);

New items will move up as required!New items will move up as required!• pq.add(10)pq.add(10)

12 17 12 24 17 81

12 17 12 24 17 81 75

10 12 12 17 17 81 75 24

Priority QueuesPriority Queues

Each element in the queue has a Each element in the queue has a prioritypriority• in our examples that’s in our examples that’s all all they havethey have• in some app.s may have other informationin some app.s may have other information

Ties can be broken in any manner at all (!)Ties can be broken in any manner at all (!)• may be FIFO, but not necessarilymay be FIFO, but not necessarily

CanCan think of it as a sorted list think of it as a sorted list• but that’s not actually what we’re going to do!but that’s not actually what we’re going to do!

Real World Priority QueuesReal World Priority Queues

Printer queuePrinter queue Time-sharing job queueTime-sharing job queue Airport baggage check with separate Airport baggage check with separate

“Executive Class” check-in“Executive Class” check-in

PQ ApplicationsPQ Applications

Asynchronous event simulationAsynchronous event simulation Sort a listSort a list

Asynchronous EventsAsynchronous Events

Simulation of timed eventsSimulation of timed events Event recordEvent record

• event type + time it occursevent type + time it occurs Keep events in a PQKeep events in a PQ

• get next event from the PQ (minimum element)get next event from the PQ (minimum element)• update the clock with the timeupdate the clock with the time• generate any new events & insert into the PQgenerate any new events & insert into the PQ

Bank SimulationBank Simulation

Customers arrive at randomCustomers arrive at random• served in the order they arriveserved in the order they arrive

» (they queue!)(they queue!)

Multiple tellers availableMultiple tellers available• serve customers as they become availableserve customers as they become available• some customers take more time than otherssome customers take more time than others

Simulate with randomnessSimulate with randomness• random arrival times, random service timesrandom arrival times, random service times

Bank Simulation EventsBank Simulation Events

Customers arrive at randomCustomers arrive at random• serve in order of arrivalserve in order of arrival• event arrival: record time of arrivalevent arrival: record time of arrival

Customers served by first available tellerCustomers served by first available teller• service takes random amount of timeservice takes random amount of time• event departure: record time service will endevent departure: record time service will end

Handle events in the order they happenHandle events in the order they happen• priority queue sorted on time of eventpriority queue sorted on time of event

Bank Simulation ParametersBank Simulation Parameters

New arrival every 1 to 10 minutesNew arrival every 1 to 10 minutes• for an hourfor an hour• can schedule these at start of simulationcan schedule these at start of simulation

Each service takes 1 to 10 minutesEach service takes 1 to 10 minutes• can’t schedule until can’t schedule until afterafter they go to the teller! they go to the teller!• events need to be added events need to be added duringduring the simulation the simulation

Calculate average & maximum wait timeCalculate average & maximum wait time

Arrival ScheduleArrival Schedule

eventPQ

arrival(4), arrival(7), arrival(8), arrival(17), …

lineUpQ

time

0

numTellersFree

2

customers

0

totalWaitingTime

0

maxWaitingTime

0

arrival(4), arrival(7), arrival(8), arrival(17), …

First Event: Arrival @ time = 4First Event: Arrival @ time = 4

eventPQ

arrival(7), arrival(8), arrival(17), arrival(18), …

lineUpQ

time

4

numTellersFree

2

customers

0

totalWaitingTime

0

maxWaitingTime

0

arrival(7), arrival(8), departure(10), arrival(17), …

1

1

arrival(7), arrival(8), departure(10), arrival(17), …

Next Event: Arrival @ 7Next Event: Arrival @ 7

eventPQ

arrival(8), departure(10), arrival(17), arrival(18), …

lineUpQ

time

7

numTellersFree

1

customers

1

totalWaitingTime

0

maxWaitingTime

0

arrival(8), departure(10), departure(11), …

0

2

arrival(8), departure(10), departure(11), …

Next Event: Arrival @ 8Next Event: Arrival @ 8

eventPQ

departure(10), departure(11), arrival(17), …

lineUpQ

time

8

numTellersFree

0

customers

2

totalWaitingTime

0

maxWaitingTime

0

arrival(8)

departure(10), departure(11), arrival(17), …

Next Event: Departure @ 10Next Event: Departure @ 10

eventPQ

departure(11), arrival(17), arrival(18), …

lineUpQ

arrival(8)

time

10

numTellersFree

0

customers

2

totalWaitingTime

0

maxWaitingTime

0

departure(11), arrival(17), departure(17), …

0

3 2 2

General Event SimulationGeneral Event Simulation

P.Q. of eventsP.Q. of events• some events added at startsome events added at start• some events added as go alongsome events added as go along• loop until P.Q. is emptyloop until P.Q. is emptyaddStartingEvents(pq);addStartingEvents(pq);while (!pq.isEmpty()) {while (!pq.isEmpty()) { Event e = pq.remove();Event e = pq.remove(); processEvent(e, pq);processEvent(e, pq);}}

Sorting with a P.Q.Sorting with a P.Q.

Add all items from array into P.Q.Add all items from array into P.Q.for (int i = 0; i < a.length; ++i) {for (int i = 0; i < a.length; ++i) { pq.add(a[i]);pq.add(a[i]);}}

Remove them in order back into the arrayRemove them in order back into the arrayfor (int i = 0; i < a.length; ++i) {for (int i = 0; i < a.length; ++i) { a[i] = pq.remove();a[i] = pq.remove();}}

That’s That’s almostalmost heap sort heap sort

QuestionsQuestions

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