Analysis and design of algorithm Unit-4

1. Sub code:10CS43 Analysis and design of algorithm Unit-4 Engineered for Tomorrow Presented by Shruthi N Asst.Professor CSE, MVJCE

2. Definition: Dynamic programming (DP) is a general algorithm design technique for solving problems with overlapping sub-problems. This technique was invented by American mathematician ―Richard Bellman‖ in 1950s. Dynamic Programming Properties • An instance is solved using the solutions for smaller instances. • The solutions for a smaller instance might be needed multiple times, so store their results in a table. • Thus each smaller instance is solved only once. • Additional space is used to save time. Dynamic Programming

3. Dynamic Programming vs. Divide & Conquer • LIKE divide & conquer, dynamic programming solves problems by combining solutions to sub-problems. UNLIKE divide & conquer, sub-problems are NOT independent in dynamic programming.

4. Warshall’s Algorithm • Directed Graph: A graph whose every edge is directed is called directed graph OR digraph • Adjacency matrix: The adjacency matrix A = {aij} of a directed graph is the boolean matrix that has • 1 - if there is a directed edge from ith vertex to the jth vertex • 0 - Otherwise • Transitive Closure: Transitive closure of a directed graph with n vertices can be defined as the n-by-n matrix T={tij}, in which the elements in the ith row (1≤ i ≤ n) and the jth column(1≤ j ≤ n) is 1 if there exists a nontrivial directed path (i.e., a directed path of a positive length) from the ith vertex to the jth vertex, otherwise tij is 0. The transitive closure provides reach ability information about a digraph.

5. Algorithm: Algorithm Warshall(A[1..n, 1..n]) // Computes transitive closure matrix // Input: Adjacency matrix A // Output: Transitive closure matrix R R(0) A for k→1 to n do for i→ 1 to n do for j→ 1 to n do R(k)[i, j]→ R(k-1)[i, j] OR (R(k-1)[i, k] AND R(k-1)[k, j] ) return R(n)

6. Example:

7. Contd.. k = 4 Vertex {1,2,3,4 } can be intermediate

8. Contd.. • Efficiency: • Time efficiency is Θ(n3) • Space efficiency: Requires extra space for separate matrices for recording intermediate results of the algorithm.

9. Single-Source Shortest Path Problem Single-Source Shortest Path Problem- The problem of finding shortest paths from a source vertex v to all other vertices in the graph.

10. Dijkstra's algorithm Dijkstra's algorithm-is a solution to the single-source shortest path problem in graph theory.   Works on both directed and undirected graphs. However, all edges must have nonnegative weights. Approach: Greedy Input: Weighted graph G={E,V} and source vertex v∈V, such that all edge weights are nonnegative Output: Lengths of shortest paths (or the shortest paths themselves) from a given source vertex v∈V to all other vertices

11. Dijkstra's algorithm - Pseudocode dist[s] ←0 (distance to source vertex is zero)for  all v ∈ V–{s}        do  dist[v] ←∞ (set all other distances to infinity) S←∅ (S, the set of visited vertices is initially empty) Q←V  (Q, the queue initially contains all vertices) while Q ≠∅ (while the queue is not empty) do   u ← mindistance(Q,dist) (select the element of Q with the min. distance)   S←S∪{u} (add u to list of visited vertices)        for all v ∈ neighbors[u]               do  if   dist[v] > dist[u] + w(u, v) (if new shortest path found)                         then      d[v] ←d[u] + w(u, v) (set new value of shortest path) (if desired, add traceback code) return dist

12. Dijkstra Animated Example

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22. Implementations and Running Times     The simplest implementation is to store vertices in an array or linked list. This will produce a running time of   O(|V|^2 + |E|) For sparse graphs, or graphs with very few edges and many nodes, it can be implemented more efficiently storing the graph in an adjacency list using a binary heap or priority queue. This will produce a running time of O((|E|+|V|) log |V|)

23. Dijkstra's Algorithm - Why It Works As with all greedy algorithms, we need to make sure that it is a correct algorithm (e.g., it always returns the right solution if it is given correct input). A formal proof would take longer than this presentation, but we can understand how the argument works intuitively.

24. Dijkstra's Algorithm - Why It Works • To understand how it works, we’ll go over the previous example again. However, we need two mathematical results first: Lemma 1: Triangle inequalityIf δ(u,v) is the shortest path length between u and v, δ(u,v) ≤ δ(u,x) + δ(x,v) • Lemma 2: The subpath of any shortest path is itself a shortest path. The key is to understand why we can claim that anytime we put a new vertex in S, we can say that we already know the shortest path to it.

25. Dijkstra's Algorithm - Why use it? • As mentioned, Dijkstra’s algorithm calculates the shortest path to every vertex. • However, it is about as computationally expensive to calculate the shortest path from vertex u to every vertex using Dijkstra’s as it is to calculate the shortest path to some particular vertex v. • Therefore, anytime we want to know the optimal path to some other vertex from a determined origin, we can use Dijkstra’s algorithm.

26. Applications of Dijkstra's Algorithm - Traffic Information Systems are most prominent use  - Mapping (Map Quest, Google Maps) - Routing Systems

27. Floyd’s Algorithm to find -ALL PAIRS SHORTEST PATHS • Weighted Graph: Each edge has a weight (associated numerical value). Edge weights may represent costs, distance/lengths, capacities, etc. depending on the problem. • Weight matrix: W(i,j) is • 0 if i=j • ∞ if no edge b/n i and j. • ―weight of edge‖ if edge b/n i and j.

28. Algorithm: Algorithm Floyd(W[1..n, 1..n]) // Implements Floyd‘s algorithm // Input: Weight matrix W // Output: Distance matrix of shortest paths‘ length D W for k → 1 to n do for i→ 1 to n do for j→ 1 to n do D [ i, j]→ min { D [ i, j], D [ i, k] + D [ k, j] return D

29. Example:

30. 0/1 Knapsack Problem Memory function • Definition: • Given a set of n items of known weights w1,…,wn and values v1,…,vn and a knapsack of capacity W, the problem is to find the most valuable subset of the items that fit into the knapsack. • Knapsack problem is an OPTIMIZATION PROBLEM

31. Dynamic programming approach to solve knapsack problem • Step 1: Identify the smaller sub-problems. If items are labeled 1..n, then a sub-problem would be to find an optimal solution for Sk = {items labeled 1, 2, .. k} • Step 2: Recursively define the value of an optimal solution in terms of solutions to smaller problems. Initial conditions: V[ 0, j ] = 0 for j ≥ 0 V[ i, 0 ] = 0 for i ≥ 0 Recursive step: max { V[ i-1, j ], vi +V[ i-1, j - wi ] } V[ i, j ] = if j - wi ≥ 0 V[ i-1, j ] if j - wi < 0 Step 3: Bottom up computation using iteration

32. Question: • Apply bottom-up dynamic programming algorithm to the following instance of the knapsack problem Capacity W= 5

33. Solution:

34. Contd..

35. Contd..

36. Travelling salesman problem • Let G(V,E) be a directed graph with edge cost cij is defined such that cij >0 for all i and j and cij = ,if <i,j>  E. • Let V =n and assume n>1. • The traveling salesman problem is to find a tour of minimum cost. • A tour of G is a directed cycle that include every vertex in V. • The cost of the tour is the sum of cost of the edges on the tour. • The tour is the shortest path that starts and ends at the same vertex

37. Application : • Suppose we have to route a postal van to pick up mail from the mail boxes located at ‘n’ different sites. • An n+1 vertex graph can be used to represent the situation. • One vertex represent the post office from which the postal van starts and return. • Edge <i,j> is assigned a cost equal to the distance from site ‘i’ to site ‘j’. • The route taken by the postal van is a tour and we are finding a tour of minimum length.

38. Example optimal cost is 35  the shortest path is, g(1,{2,3,4}) = c12 + g(2,{3,4}) => 1->2  g(2,{3,4}) = c24 + g(4,{3}) => 1->2->4  g(4,{3}) = c43 + g(3{}) => 1->2->4->3->1  so the optimal tour is 1  2  4 3  1 10 k j 15 10 15 8 6 13 9 9 20 12 l m 7

39. 1-39 Engineered for Tomorrow Assignment questions • 1. What is dynamic programming? • 2. Write an all pair shortest path problem algorithm. • 3. Write an algorithm for warshall‟s problem algorithm • 4. What are memory functions? Explain how they are used to solve the knapsack problem. Solve the instance of the knapsack problem below. Capacity W= 5 . • Item Weight Value • 1 2 $12 • 2 1 $10 • 3 3 $20 • 4 2 $15

40. 1-40 Engineered for Tomorrow Assignment questions • 6. Using Warshall‟s algorithm, obtain the transitive closure of the matrix given below. 0 1 0 0 0 1 0 0 R= 0 0 0 0 0 1 0 1

41. 1-41 Engineered for Tomorrow Assignment questions • 8. Using the Dynamic programming .solve the following Knapsack instance: • n =3,[w1,w2,w3] =[1,2,2] and [p1,p2,p3]=[18,16,6] and M=4 • 9. Write the algorithm to find the shortest path using Floyd‟s approach • 10. Explain dynamic programming. List out the differences between divide and conquer and dynamic programming.

42. Thank You..