CMPT 438 Algorithms

Why Study Algorithms?• Necessary in any computer programming problem

▫ Improve algorithm efficiency: run faster, process more data, do something that would otherwise be impossible

▫ Solve problems of significantly large size▫ Technology only improves things by a constant factor

• Compare algorithms• Algorithms as a field of study

▫ Learn about a standard set of algorithms▫ New discoveries arise▫ Numerous application areas

• Learn techniques of algorithm design and analysis

What are Algorithms?•An algorithm is a sequence of

computational steps that transform the input into the output.

•An algorithm is also a tool for solving a well-specified computational problem. ▫E.g., sorting problem:

▫<31, 26, 41, 59, 58> is an instance of the sorting problem.

•An algorithm is correct if, for every input instance, it halts with the correct output.

Analyzing Algorithms• Predict the amount of resources required:

▫ memory: how much space is needed?▫ computational time: how fast the algorithm runs?

• FACT: running time grows with the size of the input• Input size (number of elements in the input)

▫ Size of an array, # of elements in a matrix, # of bits in the binary representation of the input, vertices and edges in a graph

• Def: Running time = the number of primitive operations (steps) executed before termination▫ Arithmetic operations (+, -, *), data movement, control,

decision making (if, while), comparison

Algorithm Efficiency vs. Speed•E.g.: sorting n numbers (n = 106)

▫Friend’s computer = 109 instructions/second

▫Friend’s algorithm = 2n2 instructions (insertion sort)

▫Your computer = 107 instructions/second▫Your algorithm = 50nlgn instructions (merge sort)

Algorithm Efficiency vs. Speed•To sort 100 million numbers:•Insertion sort takes more than 23 days•Merge sort takes under 4 hours

Typical Running Time Functions• 1 (constant running time):

▫ Instructions are executed once or a few times• logN (logarithmic)

▫A big problem is solved by cutting the original problem in smaller sizes, by a constant fraction at each step

• N (linear)▫A small amount of processing is done on each input

element• N logN

▫A problem is solved by dividing it into smaller problems, solving them independently and combining the solution

Typical Running Time Functions•N2 (quadratic)

▫Typical for algorithms that process all pairs of data items (double nested loops)

•N3 (cubic)▫Processing of triples of data (triple nested

loops)•NK (polynomial)•2N (exponential)

▫Few exponential algorithms are appropriate for practical use

Why Faster Algorithms?

Insertion Sort•Idea: like sorting a hand of playing cards

▫Remove one card at a time from the table, and insert it into the correct position in the left hand compare it with each of the cards already in

the hand, from right to left

Example of insertion sort 5 2 4 6 1 3

i

A:

sortedkey

nj1

INSERTION-SORT (A, n) ⊳ A[1 . . n] for j ←2 to n do key ← A[ j] i ← j –1 while i > 0 and A[i] > key do A[i+1] ← A[i] i ← i –1 A[i+1] = keyInsertion sort sorts the elements in place.

Analysis of Insertion Sort

Analysis of Insertion Sort

Running time•Parameterize the running time by the size

of the input, since short sequences are easier to sort than long ones.

Kinds of analysesWorst-case:

• T(n) =maximum time of algorithm on any input of size n.Average-case: • T(n) =expected time of algorithm over all

inputs of size n. • Need assumption of statistical distribution of

inputs.Best-case: • Cheat with a slow algorithm that works fast on some input.

“Asymptotic Analysis”

Machine-independent timeWhat is insertion sort’s worst-case time?

•It depends on the speed of our computerBIG IDEA:•Ignore machine-dependent constants.•Look at growth of T(n) as n→∞.

Θ-notationMath:Θ(g(n)) = { f (n): there exist positive constants c1, c2, and n0

such that 0 ≤c1g(n) ≤f (n) ≤c2g(n)

for all n≥n0}

Θ-notationEngineering:•Drop low-order terms; ignore leading

constants.•Example: 3n3 + 90n2–5n+ 6046 = Θ(n3)

Best Case Analysis

Best Case Analysis•The array is already sorted

▫A[i] ≤ key upon the first time the while loop test is run (when i = j -1)

▫tj = 1

Worst Case Analysis

Worst Case Analysis•The array is in reverse sorted order

▫Always A[i] > key in while loop test▫Have to compare key with all elements to

the left of the j-th position ▫compare with j-1 elements ▫tj = j

Average Case?•All permutations equally likely.

Insertion Sort Summary•Advantages

▫Good running time for “almost sorted” arrays θ(n)

•Disadvantages▫θ(n2) running time in worst and average

caseIs insertion sort a fast sorting algorithm?•Moderately so, for small n.•Not at all, for large n.

Worst-Case and Average-Case•We usually concentrate on finding only

the worst-case running time▫an upper bound on the running time▫For some algorithms, the worst case occurs

often. E.g., searching when information is not

present in the DB▫The average case is often as bad as the

worst case.

Merge Sort

Merge SortMERGE-SORT A[1 . . n] 1.If n= 1, done. 2.Recursively sort A[ 1 . . .n/2]and A[ [n/2]+1 . . n ] . 3.“Merge” the 2 sorted lists.

Example

Divide-and-Conquer•Divide the problem into a number of

subproblems▫Similar sub-problems of smaller size

•Conquer the sub-problems▫Solve the sub-problems recursively▫Sub-problem size small enough to solve the

problems in straightforward manner•Combine the solutions to the sub-problems

▫Obtain the solution for the original problem

Merge Sort Approach• To sort an array A[p . . r]:• Divide

▫Divide the n-element sequence to be sorted into two subsequences of n/2 elements each

• Conquer▫Sort the subsequences recursively using merge

sort ▫When the size of the sequences is 1 there is

nothing more to do• Combine

▫Merge the two sorted subsequences

Merge sort

Merge sort

Analyzing merge sortMERGE-SORT A[1 . . n]1.If n= 1, done.2.Recursively sort A[ 1 . . 「 n/2 」 ] and A[ 「 n/2 」 +1 . . n ] .3.“Merge”the 2sorted lists

Sloppiness: Should be T( 「 n/2 」 ) + T( 「 n/2 」 ) , but it turns out not to matter asymptotically.

T(n)Θ(1)2T(n/2)

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Merging two sorted arrays20 1213

11 7 9 2 1

Merging two sorted arrays20

1213

11 7

9 2

1

20 1213 11 7 9 2

20 1213 11 7 9

20 1213 11 9

20 1213 11

20 1213

1 2 7 9 11 12

Time?In place sort?

Merging two sorted arrays20

1213

11 7

9 2

1

20 1213 11 7 9 2

20 1213 11 7 9

20 1213 11 9

20 1213 11

20 1213

1 2 7 9 11 12

Time = Θ(n) to merge a total of n elements (linear time).

Analyzing Divide and Conquer Algorithms•T(n) = aT(n/b) + D(n) + C(n) •The recurrence is based on the three

steps of the paradigm:▫T(n) – running time on a problem of size n▫Divide the problem into a subproblems,

each of size n/b: takes D(n)▫Conquer (solve) the subproblems: takes

aT(n/b)▫Combine the solutions: takes C(n)

MERGE – SORT Running Time•T(n) = 2T(n/2) + θ(n) if n > 1•Divide:

▫compute q as the average of p and r: D(n) = θ(1)

•Conquer:▫recursively solve 2 subproblems, each of

size n/2 -> 2T (n/2)•Combine:

▫MERGE on an n-element subarray takes θ(n) time C(n) = θ(n)

Recurrence for merge sortΘ(1) if n= 1;2T(n/2)+ Θ(n) if n> 1.

T(n) =

Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is

constant.

Conclusions• Θ(n lg n) is better than Θ(n2).• Therefore, merge sort asymptotically

beats insertion sort in the worst case.•Disadvantage

▫Requires extra space Θ (n)

Divide-and-Conquer Example:Binary SearchFind an element in a sorted array:1. Divide: Check middle element.2. Conquer: Recursively search 1 subarray.3. Combine: Trivial.

A[8] = {1, 2, 3, 4, 5, 7, 9, 11}Find 7

• For an ordered array A, finds if x is in the array A[lo…hi]

Analysis of Binary Search

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Divide-and-Conquer Example:Powering a Number

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•Homework 1•Quiz