Greedy Algorithms Pasi Fränti

Greedy algorithm 1.Coin problem 2.Minimum spanning tree 3.Generalized knapsack problem 4.Traveling salesman problem

Task: Given a coin set, pay the required amount (36 snt) using least number of coins. Coin problem

Coin problem Another coin set Amount to be paid: 30 Greedy: Optimal:

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Minimum spanning tree When greedy works Needs problem definition! Tree = … (no cycles) Spanning Tree = … Minimum = … (couple of examples with simple graph)

Prim(V, E): RETURN T Select (u,v)  E with min weight S  S  {u,v}; P  P  {(u,v)}; E  E\{(u,v)}; REPEAT Select (u,v) with min weight  (u,v)  E, u  S, v  S S  S  {v}; P  P  {(u,v)}; E  E\{(u,v)}; UNTIL S=V Return P; Minimum spanning tree Prim’s algorithm

Example of Prim

Proof of optimality General properties of spanning trees Spanning tree includes N-1 links There are no cycles Minimum spanning tree is the one with the smallest weights A B C A B C  Remove Link BC Cycle No cycle

Proof of optimality Case: minimum link 2 A B C A B C 2 1  Link AB is minimum Suppose it is not in MST Path A → B must exist Add AB Adding AB we can remove another link (e.g. AC) Path A → C exists All nodes reached from C can now be reached from B

Proof of optimality Induction step A B C 4 3 E D  Replace CD by CE MST solved for Subset S Suppose CE is minimum connecting S outside Path D → E must exist and is outside S

Proof of optimality Induction step A B C 4 3 E D MST solved for Subset S Path D → E still exist as before All nodes reachable via D can now be reached via C → D

Source of data just for fun

1. A   // initially A is empty 2. for each vertex v  V[G] // line 2-3 takes O(V) time 3. do Create-Set(v)// create set for each vertex 4. sort the edges of E by nondecreasing weight w 5. for each edge (u,v)  E, in order by nondecreasing weight 6. do if Find-Set(u)  Find-Set(v) // u&v on different trees 7. then A  A  {(u,v)} 8. Union(u,v) 9. return A Total running time is O(E lg E). Minimum spanning tree Kruskal’s algorithm Needs revisions: -Remove numbers -Change terminology

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Cardiff Sheffield Nottingham Oxford Southampton Bristol Shrewsbury Liverpool Aberystwyth B/ham Manchester

Input:Weight of N items {w 1, w 2,..., w n } Cost of N items {c 1, c 2,..., c n } Knapsack limit S Output:Selection for knapsack: {x 1,x 2,…x n } where x i  {0,1}. Sample input: w i ={1,1,2,4,12} c i = {1,2,2,10,4} S=15 Generalized Knapsack problem Problem definition

Will appear 2014… Generalized Knapsack problem

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Traveling salesman problem GreedyTSP(V, E, home): RETURN T X[1]  home; FOR i  1 TO N-1 DO Select (u,v) with min weight  (u,v)  E, u  S, v  S X[………….. S  S  {v}; …. UNTIL V≠{} Return something; Needs to be done

Greedy algorithms 1.Coin problem 2.Minimum spanning tree 3.Generalized knapsack problem 4.Traveling salesman problem Solved Not Solved Not