CS4800: Algorithms & Data Jonathan Ullman

Transcription

1 CS4800: Algorithms & Data Jonathan Ullman Lecture 13: Shortest Paths: Dijkstra s Algorithm, Heaps DFS(?) Feb 0, 018

2 Navigation s t

3 Weighted Graphs A graph with edge lengths! = #, %, {' ( } # is the set of nodes/vertices % # # is the set of edges ' ( R are edge weights Can be directed or undirected Today: Directed graphs (models one-way streets) Non-negative edge lengths denoted l ( 0

4 Shortest Paths The length of a path = is the sum of the edge lengths Shortest Path: given nodes 9, : #, find the shortest path from 9 to : Single-Source Shortest Paths: given a node 9 #, find the shortest paths from 9 to every : #

5 Structure of Shortest Paths

6 Last Week on CS4800 If all edge lengths are 1 then BFS solves SSSP

7 Last Week on CS4800 Theorem: if all edge lengths are 1, then layer ; is exactly the set of nodes at distance ; from 9

8 Extending to Edge Lengths Alternative interpretation of BFS: Maintain a set of explored nodes < Invariant: when we explore a node, we know its distance Iteration: explore the nodes at smallest distance first Can we generalize this to graphs with edge lengths?

9 Dijkstra s Algorithm (Informal) Maintain a set of explored nodes < Initially < is empty Maintain upper bound on distances = 3 Initially = 9 = 0, 3 9, = 3 = Until all nodes are explored: Choose the node A < that minimizes = A Update the distance of all neighbors of A Invariant: if A <, then = A is the length of the shortest path from 9 to A We ll talk about finding the shortest paths in a bit

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11 Dijkstra s Algorithm: Demo Graph with nonnegative edge lengths: A 10 B D C E

12 Dijkstra s Algorithm: Demo Initialize: 10 B D 0 A Q: d(v): A B C D E 0 3 C E S: {}

13 Dijkstra s Algorithm: Demo Closest node is A: 10 B D 0 A d(v): A B C D E 0 3 C E S: { A }

14 Dijkstra s Algorithm: Demo Explore edges leaving A: 0 A B D Q: d(v): A B C D E C S: { A } E 3

15 Dijkstra s Algorithm: Demo Closest node is C: B D 0 A Q: d(v): A B C D E C S: { A, C } E 3

16 Dijkstra s Algorithm: Demo Explore edges leaving C: 0 A 10 7 B 11 D Q: d(v): A B C D E C S: { A, C } E 3 5

17 Dijkstra s Algorithm: Demo Closest node is E: 10 7 B 11 D 0 A Q: d(v): A B C D E C S: { A, C, E } E 3 5

18 Dijkstra s Algorithm: Demo Explore edges leaving E: 0 A 10 7 B 11 D Q: d(v): A B C D E C S: { A, C, E } E 3 5

19 Dijkstra s Algorithm: Demo Closest node is B: 10 7 B 11 D 0 A Q: d(v): A B C D E C S: { A, C, E, B } E 3 5

20 Dijkstra s Algorithm: Demo Explore edges leaving B: 0 A 10 7 B 9 D Q: d(v): A B C D E C S: { A, C, E, B } E 3 5

21 Dijkstra s Algorithm: Demo Closest node is D: 10 7 B 9 D 0 A Q: d(v): A B C D E C E 3 5 S: { A, C, E, B, D }

22 Dijkstra s Algorithm (Informal) Invariant: if A <, then = A is the length of the shortest path from 9 to A Proof by induction:

23 Dijkstra s Algorithm (Informal) Invariant: if A <, then = A is the length of the shortest path from 9 to A Proof by induction: D C 9 < A 3

24 Dijkstra Implementation Dijkstra(G,s): Set = 9 0, set = 3 for 3 9 Set <, H # While (H is not empty): Let 3 argmin = ' O P Remove 3 from H add it to < For (neighbors 3, A %): If (= A > = 3 + l S,T ): = A = 3 + l S,T

25 Dijkstra Implementation Dijkstra(G,s): Set = 9 0, set = 3 for 3 9 Set <, H # While (H is not empty): Let 3 argmin = ' O P Remove 3 from H add it to < For (neighbors 3, A %): If (= A > = 3 + l S,T ): = A = 3 + l S,T

26 Implementing Dijkstra Every iteration we need to: Find 3 = argmin O U = ' to explore Find all neighbors 3, ' % and update =

27 Priority Queues (Heaps)

28 Priority Queues Want a data structure H to store key-value pairs (k,v(k)) efficiently: Operations: Insert(Q,k,v) ExtractMin(Q) DecreaseKey(Q,k,v)

29 Possible Approaches?

30 Heaps Store keys in a binary tree (can actually be an array) Heap Order: If V is the parent of V, 3 V 3 V X

31 Implementing ExtractMin

32 Implementing ExtractMin (HeapifyDown)

33 Implementing Insert

34 Implementing Insert (HeapifyUp)

35 Implementing DecreaseKey

36 Implementing Insert (HeapifyUp)

37 Implementation Using Arrays Maintain an array Y representing the nodes Maintain an array mapping keys to nodes Can look up the value of a given key in Z 1 time Index Arithmetic: for any node ; LeftChild ; = ; RightChild ; = ; + 1 Parent ; = ;

38 Heaps Summary Heapify operations take Z logi time for i keys ExtractMin in Z log i : Eliminate the root, move the last element to the root, restore the heap property using HeapifyDown Insert in Z log i : Put new key in the last position, restore the heap property using HeapifyUp DecreaseKey in Z log i : Lookup the location of the key, change its value, restore the heap property using HeapifyUp

39 Back to Dijkstra s Algorithm

40 Dijkstra Implementation Dijkstra(G,s): Set = 9 0, set = 3 for 3 9 Set <, H # While (H is not empty): Let 3 argmin = 3 S P Remove 3 from H add it to < For (neighbors 3, A %): If (= A > = 3 + l S,T ): = A = 3 + l S,T

41 Dijkstra Summary Can find the distance from 9 to all other nodes 3 # in time Z j log i Use a heap to implement the priority queue Can get Z j + i log i using fancier priority queues How do we find the shortest paths themselves?

42 Dijkstra Implementation Dijkstra(G,s): Set = 9 0, set = 3 for 3 9 Set last 3 Set <, H # While (H is not empty): Let 3 argmin = 3 S P Remove 3 from H add it to < For (neighbors 3, A %): If (= A > = 3 + l S,T ): = A = 3 + l S,T last A 3

43 Dijkstra s Algorithm: Demo A 10 B D C E