Chapter 4. Greedy Algorithms. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.

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1 Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright Pearson-Addison Wesley. All rights reserved. 4

2 4.1 Interval Scheduling

3 Interval Scheduling Interval scheduling. Job j starts at s j and finishes at f j. Two jobs compatible if they don't overlap. Goal: find maximum subset of mutually compatible jobs. a b c d e f g h Time

4 Interval Scheduling: Greedy Algorithms Greedy template. Consider jobs in some order. Take each job provided it's compatible with the ones already taken. [Earliest start time] Consider jobs in ascending order of start time s j. [Earliest finish time] Consider jobs in ascending order of finish time f j. [Shortest interval] Consider jobs in ascending order of interval length f j - s j. [Fewest conflicts] For each job, count the number of conflicting jobs c j. Schedule in ascending order of conflicts c j. 7

5 Interval Scheduling: Greedy Algorithms Greedy template. Consider jobs in some order. Take each job provided it's compatible with the ones already taken. breaks earliest start time breaks shortest interval breaks fewest conflicts 8

6 Interval Scheduling: Greedy Algorithm Greedy algorithm. Consider jobs in increasing order of finish time. Take each job provided it's compatible with the ones already taken. Sort jobs by finish times so that f 1 f... f n. jobs selected A for j = 1 to n { if (job j compatible with A) A A {j} } return A Implementation. O(n log n). Remember job j* that was added last to A. Job j is compatible with A if s j f j*.

7 Interval Scheduling B C A E D F G H Time

8 Interval Scheduling B C A E D F G H Time B

9 Interval Scheduling B C A E D F G H Time B C

10 Interval Scheduling B C A E D F G H Time B A

11 Interval Scheduling B C A E D F G H Time B E

12 Interval Scheduling B C A E D F G H Time B ED

13 Interval Scheduling B C A E D F G H Time B E F

14 Interval Scheduling B C A E D F G H Time B E G

15 Interval Scheduling B C A E D F G H Time B E H

16 4.1 Interval Partitioning

17 Interval Partitioning Interval partitioning. Lecture j starts at s j and finishes at f j. Goal: find minimum number of classrooms to schedule all lectures so that no two occur at the same time in the same room. Ex: This schedule uses 4 classrooms to schedule 1 lectures. e j c d g b h a f i :3 1 1: :3 1 1:3 1 1:3 :3 3 3:3 4 4:3 Time 7

18 Interval Partitioning Interval partitioning. Lecture j starts at s j and finishes at f j. Goal: find minimum number of classrooms to schedule all lectures so that no two occur at the same time in the same room. Ex: This schedule uses only 3. c d f j b g i a e h :3 1 1: :3 1 1:3 1 1:3 :3 3 3:3 4 4:3 Time 71

19 Interval Partitioning: Lower Bound on Optimal Solution Def. The depth of a set of open intervals is the maximum number that contain any given time. Key observation. Number of classrooms needed depth. Ex: Depth of schedule below = 3 schedule below is optimal. a, b, c all contain :3 Q. Does there always exist a schedule equal to depth of intervals? c d f j b g i a e h :3 1 1: :3 1 1:3 1 1:3 :3 3 3:3 4 4:3 Time 7

20 Interval Partitioning: Greedy Algorithm Greedy algorithm. Consider lectures in increasing order of start time: assign lecture to any compatible classroom. Sort intervals by starting time so that s 1 s... s n. d number of allocated classrooms for j = 1 to n { if (lecture j is compatible with some classroom k) schedule lecture j in classroom k else allocate a new classroom d + 1 schedule lecture j in classroom d + 1 d d + 1 } Implementation. O(n log n). For each classroom k, maintain the finish time of the last job added. Keep the classrooms in a priority queue. 73

21 Interval Partitioning: Greedy Analysis Observation. Greedy algorithm never schedules two incompatible lectures in the same classroom. Theorem. Greedy algorithm is optimal. Pf. Let d = number of classrooms that the greedy algorithm allocates. Classroom d was allocated because we needed to schedule a job, say j, that is incompatible with all d-1 other classrooms. Since we sorted by start time, all these incompatibilities are caused by lectures that start no later than s j. Thus, we have d lectures overlapping at time s j + ε. Key observation all schedules use d classrooms. 74

22 Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright Pearson-Addison Wesley. All rights reserved. 7

23 4.4 Shortest Paths in a Graph shortest path from Princeton CS department to Einstein's house

24 Shortest Path Problem Shortest path network. Directed graph G = (V, E). Source s, target t. Length l e = length of edge e. Shortest path problem: find shortest directed path from s to t. cost of path = sum of edge costs in path 3 3 s Cost of path s--3--t = = t 77

25 Dijkstra's Algorithm Dijkstra's algorithm. Maintain a set of explored nodes S for which we have determined the shortest path distance d(u) from s to u. Initialize S = { s }, d(s) =. Repeatedly (greedily) choose unexplored node v S which minimizes add v to S, and set d(v) = (v). shortest path to some u in explored part, followed by a single edge (u, v) S d(u) u l e v s 78

26 Dijkstra's Shortest Path Algorithm Find shortest path from s to t. 4 3 s t 7

27 Dijkstra's Shortest Path Algorithm S = { } PQ = { s,, 3, 4,,, 7, t } s distance label t

28 Dijkstra's Shortest Path Algorithm S = { } PQ = { s,, 3, 4,,, 7, t } delmin s distance label t

29 Dijkstra's Shortest Path Algorithm decrease key S = { s } PQ = {, 3, 4,,, 7, t } X s 1 X distance label 7 X t

30 Dijkstra's Shortest Path Algorithm S = { s } PQ = {, 3, 4,,, 7, t } X delmin s 1 X distance label 7 X t

31 Dijkstra's Shortest Path Algorithm S = { s, } PQ = { 3, 4,,, 7, t } X s 1 X X t

32 Dijkstra's Shortest Path Algorithm S = { s, } PQ = { 3, 4,,, 7, t } decrease key X X 33 s 1 X X t

33 Dijkstra's Shortest Path Algorithm S = { s, } PQ = { 3, 4,,, 7, t } X X 33 s 1 X delmin X t

34 Dijkstra's Shortest Path Algorithm S = { s,, } PQ = { 3, 4,, 7, t } 3 X X 33 X s 1 X 3 44 X X t

35 Dijkstra's Shortest Path Algorithm S = { s,, } PQ = { 3, 4,, 7, t } 3 X X 33X s 1 X 3 44 X X 1 delmin t

36 Dijkstra's Shortest Path Algorithm S = { s,,, 7 } PQ = { 3, 4,, t } 3 X X 33X s 1 X 3 44 X X X 1 44 t X 8

37 Dijkstra's Shortest Path Algorithm S = { s,,, 7 } PQ = { 3, 4,, t } delmin 3 X X 33X s 1 X 3 44 X X X 1 44 t X

38 Dijkstra's Shortest Path Algorithm S = { s,, 3,, 7 } PQ = { 4,, t } 3 X X 33X s 1 X 3 44 X X X X t X X 1

39 Dijkstra's Shortest Path Algorithm S = { s,, 3,, 7 } PQ = { 4,, t } 3 X X 33X s 1 X X 3 X 34 X delmin X t X X

40 Dijkstra's Shortest Path Algorithm S = { s,, 3,,, 7 } PQ = { 4, t } 3 X X 33X s 1 X 3 44 X X X X X X t X X 3

41 Dijkstra's Shortest Path Algorithm S = { s,, 3,,, 7 } PQ = { 4, t } 3 X X 33X s 1 X 3 44 X X X X 4 delmin X X t X X 4

42 Dijkstra's Shortest Path Algorithm S = { s,, 3, 4,,, 7 } PQ = { t } 3 X X 33X s 1 X 3 44 X X X X X X t X X

43 Dijkstra's Shortest Path Algorithm S = { s,, 3, 4,,, 7 } PQ = { t } 3 X X 33X s 1 X 3 44 X X X X X 1 44 delmin 1 X t X X

44 Dijkstra's Shortest Path Algorithm S = { s,, 3, 4,,, 7, t } PQ = { } 3 X X 33X s 1 X 3 44 X X X X X X t X X 7

45 Dijkstra's Shortest Path Algorithm S = { s,, 3, 4,,, 7, t } PQ = { } 3 X X 33X s 1 X 3 44 X X X X X X t X X 8

46 Dijkstra's Algorithm: Proof of Correctness Invariant. For each node u S, d(u) is the length of the shortest s-u path. Pf. (by induction on S ) Base case: S = 1 is trivial. Inductive hypothesis: Assume true for S = k 1. Let v be next node added to S, and let (u,v) be the chosen edge. The shortest s-u path plus (u, v) is an s-v path of length (v). Consider any s-v path P. We'll see that it's no shorter than (v). P Let x-y be the first edge in P that leaves S, and let P' be the subpath to x. P' x P is already too long as soon as it s leaves S. S u l (P) l (P') + l (x,y) d(x) + l (x, y) (y) (v) v y nonnegative weights inductive hypothesis defn of π(y) Dijkstra chose v instead of y

47 Dijkstra's Algorithm: Implementation For each unexplored node v S, explicitly maintain Next node to explore = node v S with minimum (v). When exploring v, for each incident edge e = (v, w), w S, update Efficient implementation. Maintain a priority queue of unexplored nodes, prioritized by (v). PQ Operation Insert ExtractMin ChangeKey n n 1 log n log n log n Priority Queue IsEmpty n Total (w) = min { (w), (v)+ l e }. Dijkstra n n m Array Binary heap d-way Heap Fib heap n m log n d log d n d log d n log d n m log m/n n 1 log n 1 m + n log n Individual ops are amortized bounds 1

48 Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright Pearson-Addison Wesley. All rights reserved. 11