Fundamental Techniques

week 4 Complexity of Algorithms 1

Fundamental Techniques

There are some algorithmic tools that are quite specialised. They are good for problems they are intended to solve, but they are not very versatile.There are also more fundamental (general) algorithmic tools that can be applied to a wide variety of different data structure and algorithm design problems.


The Greedy Method

An optimisation problem (OP) is a problem that involves searching through a set of configurations to find one that minimises or maximizes an objective function defined on these configurationsThe greedy method solves a given OP going through a sequence of (feasible) choicesThe sequence starts from well-understood starting configuration, and then iteratively makes the decision that seems best from all those that are currently possible.


The Greedy Method

The greedy approach does not always lead to an optimal solution.The problems that have a greedy solution are said to posses the greedy-choice property.The greedy approach is also used in the context of hard (difficult to solve) problems in order to generate an approximate solution.


Fractional Knapsack Problem

In fractional knapsack problem, where we are given a set S of n items, s.t., each item I has a positive benefit bi and a positive weight wi, and we wish to find the maximum-benefit subset that doesn’t exceed a given weight W.We are also allowed to to take arbitrary fractions of each item.



I.e., we can take an amount xi of each item i such that

The total benefit of the items taken is determined by the objective function





In the solution we use a heap-based PQ to store the items of S, where the key of each item is its value indexWith PQ, each greedy choice, which removes an item with the greatest value index, takes O(log n) timeThe fractional knapsack algorithm can be implemented in time O(n log n).



Fractional knapsack problem satisfies the greedy-choice property, henceThm: Given an instance of a fractional knapsack problem with set S of n items, we can construct a maximum benefit subset of S, allowing for fractional amounts, that has a total weight W in O(n log n) time.


Task Scheduling

Suppose we are given a set T of n tasks, s.t., each task i has a start time si and a completion time fi.Each task has to be performed on a machine and each machine can execute only one task at a time.Two tasks i and j are non-conflicting if fi sj or fj si.Two tasks can be executed on the same machine only if they are non-conflicting.


Task Scheduling

The task scheduling problem is to schedule all the tasks in T on the fewest machines possible in a non-conflicting way


Task Scheduling (algorithm)


Task Scheduling (analysis)

In the algorithm TaskSchedule, we begin with no machines and we consider the tasks in a greedy fashion, ordered by their start times.For each task i, if we have the machine that can handle task i, then we schedule i on that machine.Otherwise, we allocate a new machine, schedule i on it, and repeat this greedy selection process until we have considered all the tasks in T.


Task Scheduling (analysis)

Task scheduling problem satisfies the greedy-choice property, henceThm: Given an instance of a task scheduling problem with set of n tasks, the algorithm TaskSchedule produces a schedule of the tasks with the minimum number of machines in O(n log n) time.


Divide and Conquer

Divide: if the input size is small then solve the problem directly; otherwise divide the input data into two or more disjoint subsetsRecur: recursively solve the sub-problems associated with the subsetsConquer: take the solutions to the sub-problems and merge them into a solution to the original problem


Divide and Conquer

To analyse the running time of a divide-and-conquer algorithm we utilise a recurrence equation, whereT(n) denotes the running time of the algorithm on an input of size n, andCharacterise T(n) using an equation that relates T(n) to values of function T for problem sizes smaller than n, e.g.,


Substitution Method

One way to solve a divide-and-conquer recurrence equation is to use the iterative substitution method, a.k.a., plug-and-chug method, e.g., having

We get

And after i-1 substitutions we have

And for i = log n, we get


Recursion Tree (visual approach)

In recursion tree method, some overhead (forming a part of a recurrence equation) is associated with every node of the tree. E.g., having

Where the overhead corresponds to summand +bn. We get

The value of T(n) corresponds to the sum of all overheads. In this example, depth of the tree times overhead at each level, which is O(n log n)


Guess-and-Prove

In guess-and-prove method the solution to a recurrence equation is guessed and then proved by mathematical induction

We guess that T(n) = O(n log n). We have to prove that T(n) < C n· log n for some constant C and large enough n. We use inductive assumption that T(n/2) < C · n/2 · log (n/2) = Cn/2·(log n –1) = (Cn · log n)/2 – Cn/2.T(n) = 2T(n/2) +bn < 2((Cn · log n)/2 – Cn/2) +bn = Cn · log n + (-Cn + bn) < Cn · log n, for any C > b.


The Master Method


Matrix Multiplication

Suppose we are given two n x n matrices X and Y, and we wish to compute their product Z=X·Y, which is defined so that:

Which naturally leads to a simple O(n3) time algorithm.


Matrix Multiplication

Another way of viewing this product is in terms of sub-matrices:

where

However this gives a divide-and-conquer algorithm with running time T(n), s.t., T(n) =8T(n/2) +bn2 = O(n3)


Strassen’s Algorithm

Define seven matrix products:



Having Sis we can represent I, J, K, L:



Thus, we can compute Z=XY using seven recursive multiplications of matrices of size (n/2) x (n/2), where

One can prove, e.g., using Master Theorem, that: Thm: We can multiply two n x n matrices in O(nlog 7) = O(n2.808) time.


Dynamic Programming

The dynamic programming (DP) algorithm-design technique is similar to divide-and-conquer technique.The main difference is in replacing (possibly) repetitive recursive calls by the reference to already computed values stored in a special table.


Dynamic Programming

DP technique is used primarily for optimisation problemsWe very often apply DP where the brute-force search for the best is infeasibleHowever DP is efficient only if the problem has a certain amount of structure that we can exploit


Dynamic Programming

Simple sub-problems: there must be a way of braking the whole optimisation problem into smaller pieces sharing a similar structureSub-problem optimality: an optimal solution to the global problem must be a composition of optimal sub-problem solutionsSub-problem overlap: optimal solutions to unrelated sub-problems can contain sub-problems in common


0-1 Knapsack Problem

In 0-1 knapsack problem, is the knapsack problem where taking fractions of items is not allowed, i.e., each item si S, for 1 i

n, must be entirely accepted or rejectedItem si has a benefit bi (s.t., b1 b2 … bn) and an integer weight wi

We have the following objective:

where T S



Exponential solution: we can easily solve 0-1 knapsack problem in O(2n) time by testing all possible subsets of itemsUnfortunately exponential complexity is not acceptable for large n and we rather have to focus on nice characterisation for sub-problems in order to use DP approach



Let Sk = {si: i= 1,2,…,k}

Let B[k,w] be the maximum total benefit of a subset of Sk from among all those subsets having total weight exactly wWe have b[0,w]=0, for each wW, and





The running time of the 01Knapsack algorithm is dominated by the two nested for-loops, where the outer one iterates n times and the inner one iterates at most W timesThm: 01Knapsack algorithm finds the highest benefit subset of S with total weight at most W in O(nW) time

Fundamental Techniques

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greedy solution

greedy choice

task i

item i

greedy fashion

greedychoice property

set s of n items

set t of n tasks