Algorithm Design and Analysis
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1 Algorithm Design and Analysis LECTURE 7 Greedy Graph Algorithms Shortest paths Minimum Spanning Tree Sofya Raskhodnikova 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.1
2 The (Algorithm) Design Process 1. Work out the answer for some examples. Look for a general principle Does it work on *all* your examples? 3. Write pseudocode 4. Test your algorithm by hand or computer Does it work on *all* your examples? Python is a great language for testing algorithms 5. Prove your algorithm is always correct 6. Check running time Be prepared to go back to step 1! 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.
3 Writing algorithms Clear and unambiguous Test: You should be able to hand it to any student in the class, and have them convert it into working code. Homework pitfalls: remember to specify data structures (list, stack, hash table, ) For each function invocation, specify clearly what variables are passed to the function and what the function is returning. writing recursive algorithms: don t confuse the recursive subroutine with the first call label global variables clearly 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.3
4 Writing proofs State upfront the claim you are proving. Purpose Determine for yourself that algorithm is correct Convince reader Who is your audience? Yourself Your classmates Not the TA/grader Main goal: Find your own mistakes 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.4
5 Homework Goals: Reinforce and clarify material from lecture Develop your skills Problem-solving Communication Make sure you understand the solution Use the feedback If you don t understand something, ask! Me or the TA or on Piazza Do not copy from other sources 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.5
6 Shortest Paths 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.6
7 Shortest Path Problem Input: Directed graph G = (V, E). Source node s, destination node t. for each edge e, length (e) = length of e. length of a path = sum of lengths of edges on the path Find: shortest directed path from s to t. s Length of path (s,,3,5,t) is = t 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.7
8 Dijksta s Algorithm: Overview Maintain a set of explored nodes S whose shortest path distance d(u) from s to u is known. Initialize S = { s }, d(s) = 0. Repeatedly choose unexplored node v which minimizes π v = d u + l e min e= u,v :u S 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 (e) v s 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.8
9 Dijksta s Algorithm: Overview Maintain a set of explored nodes S whose shortest path distance d(u) from s to u is known. Invariant: d(u) is known Initialize S = { s }, d(s) = 0. for all vertices in S Repeatedly choose unexplored node v which minimizes π v = d u + l e min e= u,v :u S add v to S, and set d(v) = (v). Intuition: like BFS, but with weighted edges S s shortest path to some u in explored part, followed by a single edge (u, v) d(u) u (e) v 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.9
10 Correctness Proof of Dijkstra s (Greedy Stays Ahead) Invariant. For each node u S, d(u) is the length of the shortest path from s to u. Proof: (by induction on S ) Base case: S = 1; d(s)=0. Inductive hypothesis: Assume 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). Let (x,y) be the first edge in P that leaves S, and let P' be the subpath to x. P' + (x,y) has length d(x)+ (x,y) (y) (v) inductive hypothesis defn of (y) Dijkstra s chose v instead of y 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne s S P' u x P v y L7.10
11 Implementation For unexplored nodes, maintain π v = d u + l e min e= u,v :u S Next node to explore = node with minimum (v). When exploring v, for each edge e = (v,w), update π w = min{π w, π v + l e. Efficient implementation: Maintain a priority queue Q of unexplored nodes, prioritized by (v). 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.11
12 Implementation: priority queues Maintain a set of items with priorities (= keys ) Example: jobs to be performed Operations: INSERT DECREASE-KEY EXTRACT-MIN: find and remove item with least key Common data structure: heap Time: O(log n) per operation 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.1
13 Demo of Dijkstra s Algorithm Graph with nonnegative edge lengths: A 10 B D C E 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.13
14 Demo of Dijkstra s Algorithm Initialize: 10 B D 0 A Q: (v): A B C D E 0 3 C E S: {} 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.14
15 Demo of Dijkstra s Algorithm EXTRACT-MIN(Q) is A: 10 B D 0 A Q: (v): A B C D E 0 3 C E S: { A } 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.15
16 Demo of Dijkstra s Algorithm Explore edges leaving A: 0 A B D Q: (v): A B C D E C S: { A } E 3 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.16
17 Demo of Dijkstra s Algorithm EXTRACT-MIN(Q) is C: 0 A B D Q: (v): A B C D E C S: { A, C } E 3 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.17
18 Demo of Dijkstra s Algorithm Explore edges leaving C: 0 A 10 7 B 11 D Q: (v): A B C D E C S: { A, C } E 3 5 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.18
19 Demo of Dijkstra s Algorithm EXTRACT-MIN(Q) is E: 0 A 10 7 B 11 D Q: (v): A B C D E C S: { A, C, E } E 3 5 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.19
20 Demo of Dijkstra s Algorithm Explore edges leaving E: 0 A 10 7 B 11 D Q: (v): A B C D E C S: { A, C, E } E 3 5 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.0
21 Demo of Dijkstra s Algorithm EXTRACT-MIN(Q) is B: 0 A 10 7 B 11 D Q: (v): A B C D E C S: { A, C, E, B } E 3 5 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.1
22 Demo of Dijkstra s Algorithm Explore edges leaving B: 0 A 10 7 B 9 D Q: (v): A B C D E C S: { A, C, E, B } E 3 5 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.
23 Demo of Dijkstra s Algorithm EXTRACT-MIN(Q) is D: 0 A 10 7 B 9 D Q: (v): A B C D E C E 3 5 S: { A, C, E, B, D } 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.3
24 Pseudocode for Dijkstra(G, ) d[s] 0 for each v V {s} do d[v] [v] S Q V Q is a priority queue maintaining V S, keyed on [v] while Q do u EXTRACT-MIN(Q) S S {u}; d[u] [u] for each v Adjacency-list[u] do if [v] > [u] + (u, v) then [v] d[u] + (u, v) explore edges leaving v Implicit DECREASE-KEY 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.4
25 Analysis of Dijkstra n times while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] explore an edge do if d[v] > d[u] + (u, v) then d[v] d[u] + (u, v) \\ m implicit DECREASE-KEY s. PQ Operation ExtractMin DecreaseKey Dijkstra n m Array n 1 Binary heap log n log n d-way Heap HW HW Fib heap log n 1 Total n m log n m log m/n n m + n log n 9/14/016 Individual ops are amortized bounds S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.5
26 Physical intuition System of pipes filling with water Vertices are intersections Edge length = pipe length d(v) = time at which water reaches v Balls and strings Vertices balls Edge e string of length l(e) Hold ball s up in the air d v = (height of s)-(height of v) Nature uses greedy algorithms 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.6
27 Review Is Dijsktra s algorithm correct with negative edge weights? Give either a proof of correctness, or an example of a graph where Dijkstra fails 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.7
28 Further reading Erickson s lecture notes: 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.8
29 Minimum Spanning Tree 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.9
30 Minimum spanning tree (MST) Input: A connected undirected graph G = (V, E) with weight function w : E R. For now, assume all edge weights are distinct. Definition: A spanning tree is a tree that connects all vertices. Output: A spanning tree T of minimum weight: w( T ) w( u, v). ( u, v) T 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.30
31 Example of MST /14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.31
32 Example of MST /14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.3
33 Greedy Algorithms for MST Kruskal's: Start with T =. Consider edges in ascending order of weights. Insert edge e in T unless doing so would create a cycle. Reverse-Delete: Start with T = E. Consider edges in descending order of weights. Delete edge e from T unless doing so would disconnect T. Prim's: Start with some root node s. Grow a tree T from s outward. At each step, add to T the cheapest edge e with exactly one endpoint in 5. Borůvka s: Start with T =. At each round, add the cheapest edge leaving each connected component of T. 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.33
34 Cycles and Cuts Cycle: Set of edges of the form (a,b),(b,c),,(y,z),(z,a) Cycle C = (1,),(,3),(3,4),(4,5),(5,6),(6,1) 7 8 Cut: a subset of nodes S. The corresponding cutset D is the subset of edges with exactly one endpoint in S S Cut S = { 4, 5, 8 } Cutset D = (5,6), (5,7), (3,4), (3,5), (7,8) 7 8 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.34
35 Cycle-Cut Intersection Claim. A cycle and a cutset intersect in an even number of edges. Proof: A cycle has to leave and enter the cut the same number of times. C S V - S 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.35
36 Cut and Cycle Properties Cut property. Let S be a subset of nodes. Let e be the min weight edge with exactly one endpoint in S. Then the MST contains e. Cycle property. Let C be a cycle, and let f be the max weight edge in C. Then the MST does not contain f. f C S e e is in the MST f is not in the MST 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.36
37 Proof of Cut Property Cut property: Let S be a subset of nodes. Let e be the min weight edge with exactly one endpoint in S. Then the MST T* contains e. Proof: (exchange argument) Suppose e does not belong to T*. Adding e to T* creates a cycle C in T*. Edge e is both in the cycle C and in the cutset D corresponding to S there exists another edge, say f, that is in both C and D. T' = T* { e } - { f } is also a spanning tree. Since c e < c f, cost(t') < cost(t*). Contradiction. S f e T* 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.37
38 Proof of Cycle Property Cycle property: Let C be a cycle in G. Let f be the max weight edge in C. Then the MST T* does not contain f. Proof: (exchange argument) Suppose f belongs to T*. Deleting f from T* creates a cut S in T*. Edge f is both in the cycle C and in the cutset D corresponding to S there exists another edge, say e, that is in both C and D. T' = T* { e } - { f } is also a spanning tree. Since c e < c f, cost(t') < cost(t*). Contradiction. S f e T* 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.38
39 Greedy Algorithms for MST Kruskal's: Start with T =. Consider edges in ascending order of weights. Insert edge e in T unless doing so would create a cycle. Reverse-Delete: Start with T = E. Consider edges in descending order of weights. Delete edge e from T unless doing so would disconnect T. Prim's: Start with some root node s. Grow a tree T from s outward. At each step, add to T the cheapest edge e with exactly one endpoint in 5. Borůvka s: Start with T =. At each round, add the cheapest edge leaving each connected component of T. 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.39
40 Prim's Algorithm: Correctness Prim's algorithm. [Jarník 1930, Prim 1959] Apply cut property to S. When edge weights are distinct, every edge that is added must be in the MST Thus, Prim s algorithm outputs the MST S 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.40
41 Correctness of Kruskal [Kruskal, 1956]: Consider edges in ascending order of weight. Case 1: If adding e to T creates a cycle, discard e according to cycle property. Case 1 e Case : Otherwise, insert e = (u, v) v e S into T according to cut property where S = set of nodes in u's connected u component. Case 9/14/016 S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne L7.41
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