Analysis and Design of Algorithms

Deepak JohnDepartment Of Computer Applications, SJCET-Pala

Types of ProblemsTypes of Problems Optimization problem: construct a solution that

maximizes or minimizes some objective functionmaximizes or minimizes some objective function Decision problem: A question that has two possible

answers, yes and no.

Examples:•SHORTEST-PATH (optimization)

Given G, u,v, find a path from u to v with fewest edges.•PATH (decision)

Given G, u,v, and k, whether exist a path from u to vGiven G, u,v, and k, whether exist a path from u to vconsisting of at most k edges.

The class PThe class P P is the set of decision problems that can be solved in polynomial

time .if the size of the input to the problem is n then the problemp p pcan be solved in time O(nk) for some constant k.

Polynomially boundedan algorithm is said to be polynomilly bounded if its worst casecomplexity is bounded by a polynomial function of the inputi (if th i l i l h th t f h i t f isize(if there is a polynomial p such that for each input of size n

the algorithm terminates after at most p(n)steps) P: Polynomial algorithms - These include all the sorting P: Polynomial algorithms These include all the sorting

algorithms, with running times on the order of n, lg n,n lg n and n2.

Class NPClass NP Class NP: Problems that can be solved in a polynomial number

of steps by a nondeterministic polynomial algorithms.of steps by a nondeterministic polynomial algorithms. A nondeterministic algorithm: a two-stage procedure that takes

as its input an instance I of a decision problem and does thefollowing

1. “guessing” stage: An arbitrary string S is generated that canb h h f l i f h i ibe thought of as a guess at a solution for the given instance

2. “verification” stage: A deterministic algorithm takes both Iand S as its input and check if S is a solution to instance Iand S as its input and check if S is a solution to instance I,(outputs yes if s is a solution and outputs no or not halt at allotherwise)

NP-CompleateNP Compleate is the class of problems that are the hardest problems in NP. example: example:

A hamiltonian cycle of an undirected graph is a simplecycle that contains every vertex

The Relationship between P ,NP and NPC

Approximation algorithmsApproximation algorithms an algorithm which return solutions that are guaranteed to

be close to an optimal solution.be close to an optimal solution. We use performance ratio to measure the quality of an

approximation algorithm. We are going to find a Near-Optimal solution for a given

problem. We assume two hypothesis :1. Each potential solution has a positive cost.2. The problem may be either a maximization or a

minimization problem on the cost.

Performance ratio For minimization problem, the performance ratio of algorithm A is

defined as a number r such that for any instance I of the problem,

OPT(I) is the value of the optimal solution for instance I andA(I) is the value of the solution returned by algorithm A on instance IA(I) is the value of the solution returned by algorithm A on instance I.

For maximization problem, the performance ratio of algorithm A isdefined as a number r such that for any instance I of the problem,

OPT(I)/A(I)is at most r (r≥1),

Bin Packing ProblemBin Packing Problem Pack a set of N BINS = {1, 2, …, n} items, each with size ti , i=1,

2,…,n, into identical bins, each with capacity C. the bin packing problem is to determine the minimum number of bins

to accommodate all items. Fi di th ti l l ti i NP h d bl Finding the optimal solution is a NP-hard problem. Minimize the number of bins without violating the capacity constraints

Online bin packing Items arrive one by one.y Each item must be assigned immediately to a bin, without

knowledge of any future items. Reassignment is not allowed. There exists no online bin packing algorithm that always finds an

optimal solution. Next Fit (NF) First Fit (FF) Best Fit (BF) Next Fit (NF), First Fit (FF), Best Fit (BF)Offline bin packing All n items are known in advance, i.e. before they have to be packed., f y p Initially sort the items in decreasing order of size and assign the

largest items first. First Fit Decreasing (FFD) ,Best Fit Decreasing (BFD)

Consider items 3; 6; 2; 1; 5; 7; 2; 4; 1; 9: with bin size 10Next FitNext Fit Put the current item in the current bin if it fits, otherwise in the next

bin. O(n) Ex:[3; 6] - [2; 1; 5] -[7; 2]- [4; 1] - [9]First Fit Put the current item into the first bin it fits into.

O( 2) O(n2) Ex:[3; 6; 1] - [2; 5; 2; 1] - [7] - [4] - [9]

Best Fit Assign an arriving item to the bin in which it fits bestg g Performance of BF and FF is similar. Ex:[3;6;1]-[2;5;2;1]-[9]-[4]-[9]

Algorithm First-Fit1: forAll objects i = 1, 2, . . . , n do2: forAll bins j = 1, 2, . . . do3: if Object i fits in bin j then4: Pack object i in bin j4: Pack object i in bin j.5: Break the loop and pack the next object.6: end if7: end for8: if Object i did not fit in any available bin then9 C bi d k bj i9: Create new bin and pack object i.10: end if11: end for11: end for

Consider items 3; 6; 2; 1; 5; 7; 2; 4; 1; 9: with bin size 10First Fit DecreasingFirst Fit Decreasing1. Sort the objects in decreasing order2. Apply First Fit strategy to this sorted list.2. Apply First Fit strategy to this sorted list.

Ex:[9; 1] -[7; 3] -[6; 4] -[5; 2; 2; 1]Best Fit Decreasingg1. Sort the objects in decreasing order2. Apply Best Fit strategy to this sorted list.

Graph Coloring ProblemGraph Coloring Problem Graph coloring is an assignment of colors to the vertices of a graph.

no two adjacent vertices have the same color Chromatic number: is the smallest number of colors with which it

can be colored. S ti l l i d id l i Sequential coloring and widgerson coloring

Sequential Graph coloring1 Gi G (VE) ith ti U th i t {1 2 3 } t1. Given G=(V,E) with n vertices. Use the integers {1,2,3, …, n} to

represent colors. Start by assigning 0 to every vertex. Process thevertices one at a time.

2. For each vertex, Vi, start by coloring Vi with the color 1.3. Check the neighbors of Vi to see if any is colored 1. If not then go

t th t t Vi+1to the next vertex, Vi+1.4. If there is a neighbor colored 1, recolor Vi with color 2, and

repeat the neighbor searchrepeat the neighbor search.5. Repeat the previous step incrementing the color until we find a

color c that has not been used to color any of Vi’s neighbors.

Example for sequential coloring

Widgerson coloring Recursive Algorithm Recursive Algorithm A graph with maximum degree ∆ can be easily colored using ∆ +1

colors. Base Case: 2 Colorable Graphs Find the sub graph of the Neighborhood of a given vertex, recursively color this sub graph. At most 3√n colors for an n-colorable graph.

Algorithm

Travelling Salesman ProblemTravelling Salesman Problem Definition: Find a path through a weighted graph which starts and

ends at the same vertex, includes every other vertex exactly once,, y y ,and minimizes the total cost of edges.

TSP is classify as NP-complete problem, that means nol i l l ith t t ithi t blpolynomial algorithm can guarantee to come within countable

times of the shortest tour. TSP : Find a tour of minimum cost (distance)f ( ) Problem: To find a Hamiltonian cycle of minimal cost.

Nearest Neighbor1 Pick a reference vertex to start at1. Pick a reference vertex to start at.2. Walk to its nearest neighbor (i.e., along the shortest possible

edge).(If there is a tie, break it randomly.)3. At each stage in your tour, walk to the nearest neighbor that

you have not already visited.4 h h i i d ll i h i4. When you have visited all vertices, return to the starting vertex.

Shortest-Link Idea: Start in the middle Idea: Start in the middle.1. Add the cheapest available edge to your tour.(If there is a tie,

break it randomly.)2. Repeat until you have a Hamilton circuit.3. Make sure you add exactly two edges at each vertex.4. Don't close the circuit until all vertices are in it.