Algorithms and

Efficiency of Algorithms

February 4th

Today OutlineMore algorithms

• Variations of sequential searchPractical applications• Pattern matching • Data cleanup

There are many different algorithms to solve the same problem.

• So, how do we know when we have a good one???• I.e., how do we measure the efficiency of an

algorithm?

Write algorithms for

• Find all occurences of target

• Find number of occurences of target

• Find number of values larger than target

• Find largest

• Find smallest

• Find sum

• Find average

A Search Application in Bioinformatics

• Human genome: sequence of billions of nucleotides• Gene

– Determines human behavior– Sequence of tens of thousands of nucleotides{T,C,A,G}– The sequence is not fully known, only a portion of it

• Problem: How to determine a gene in the human genome?

Genome: …….TCAGGCTAATCGTAGG…….Gene probe: TAATC

Idea: Find all matches of the probe within the genome and then examine the nucleotides in that neighborhood

A Search Application in Bioinformatics

• Problem: – Suppose we have a text T = TCAGGCTAATCGTAGG and a pattern P =

TA. Design an algorithm that searches T to find the position of every instance of P that appears

• E.g., for this text, the algorithm should return the answer:There is a match at position 7

There is a match at position 13

This problem is a variation of the search algorithm, except that for every possible starting position every character of P must be compared with a character of T.

Pattern Matching

• Input– Text of n characters T1, T2, …, Tn

– Pattern of m (m < n) characters P1, P2, …Pm

• Output: – Location (index) of every occurrence of pattern within

text

• Algorithm: – What is the idea?

Pattern Matching

• Algorithm idea:– Check if pattern matches starting at position 1– Then check if it matches starting at position 2– …and so on

• How to check if pattern matches text starting at position k?– Check that every character of pattern matches corresponding

character of text

• How many loops will you need?

Pattern Matching

• Algorithm idea– Get input (text and pattern)

– Set starting location k to 1

– Repeat until reach end of text• Attempt to match every character in the pattern beginning at

pos k in text

• If there was a match, print k

• Add 1 to k

– Stop

• Question: is this an algorithm? – Yes, at a high level of abstraction

– Now we need to write in pseudocode

Pattern Matching Algorithm (Fig. 2.12)

Get values for n, m, the text T1T2…Tn and the pattern P1P2…Pm

Set k to 1Repeat until k>n-m+1

Set i to 1Set Mismatch to NORepeat until either (i>m) or (Mismatch = YES)

if Pi ≠ Tk+(i-1) then

Set Mismatch to YESelse Increment i by 1

if Mismatch = NO then Print the message “There is a match at position” k

increment k by 1Stop

Variations on the pattern matching algorithm

•Find only the first match for P in T.

•Find only the last match for P in T.

Comparing Algorithms• Algorithm

– Design– Correctness– Efficiency– Also, clarity, elegance, ease of understanding

• There are many ways to solve a problem– Conceptually– Also different ways to write pseudocode for the same

conceptual idea

• How to compare algorithms?

Efficiency of Algorithms

• Efficiency: Amount of resources used by an algorithm

• Space (number of variables)

• Time (number of instructions)

• When design algorithm must be aware of its use of resources

• If there is a choice, pick the more efficient algorithm!

Efficiency of Algorithms

Does efficiency matter?• Computers are so fast these days…

• Yes, efficiency matters a lot!– There are problems (actually a lot of them) for which

all known algorithms are so inneficient that they are impractical

– Remember the shortest-path-through-all-cities problem from Lab1…

Efficiency of Algorithms

How to measure time efficiency?

• Running time: let it run and see how long it takes– On what machine?

– On what inputs?

Time efficiency depends on input

• Example: the sequential search algorithm– In the best case, how fast can the algorithm halt?

– In the worst case, how fast can the algorithm halt?

Time Efficiency

• We want a measure of time efficiency which is independent of machine, speed etc– Look at an algorithm pseudocode and estimate its running time

– Look at 2 algorithm pseudocodes and compare them

• Efficiency of an algorithm: – the number of pseudocode instructions (steps) executed

• Is this accurate? – Not all instructions take the same amount of time…

– But..Good approximation of running time in most cases

Data Cleanup Algorithms

What are they?

A systematic strategy for removing errors from data.

Why are they important?

Errors occur in all real computing situations.

How are they related to the search algorithm?

To remove errors from a series of values, each value must be examined to determine if it is an error.

E.g., suppose we have a list d of data values, from which we want to remove all the zeroes (they mark errors), and pack the good values to the left. Legit is the number of good values remaining when we are done.

d1 d2 d3 d4 d5 d6 d7 d8

5 3 4 0 6 2 4 0Legit

Data Cleanup: Copy-Over algorithm

Idea: Scan the list from left to right and copy non-zero values to a new list

Copy-Over Algorithm (Fig 3.2)• Get values for n and the list of n values A1, A2, …, An• Set left to 1• Set newposition to 1• While left <= n do

• If Aleft is non-zero • Copy A left into B newposition

(Copy it into position newposition in new list• Increase left by 1• Increase newposition by 1

• Else increase left by 1 • Stop

Data Cleanup: The Shuffle-Left Algorithm

• Idea: – go over the list from left to right. Every time we see a

zero, shift all subsequent elements one position to the left.

– Keep track of nb of legitimate (non-zero) entries

• How does this work?

• How many loops do we need?

Shuffle-Left Algorithm (Fig 3.1)

1 Get values for n and the list of n values A1, A2, …, An2 Set legit to n3 Set left to 14 Set right to 25 Repeat steps 6-14 until left > legit

6 if Aleftt ≠ 0

7 Increase left by 1 8 Increase right by 1

9 else10 Reduce legit by 111 Repeat 12-13 until right > n

12 Copy Aight into Aright-1

13 Increase right by 114 Set right to left + 1

15 Stop

Exercising the Shuffle-Left Algorithm

d1 d2 d3 d4 d5 d6 d7 d8

5 3 4 0 6 2 4 0legit

Data Cleanup: The Converging-Pointers Algorithm

• Idea:– One finger moving left to right, one moving

right to left– Move left finger over non-zero values;– If encounter a zero value then

• Copy element at right finger into this position

• Shift right finger to the left

Converging Pointers Algorithm (Fig 3.3)

1 Get values for n and the list of n values A1, A2,…,An

2 Set legit to n3 Set left to 14 Set right to n5 Repeat steps 6-10 until left ≥ right

6 If the value of Aleft≠0 then increase left by 1

7 Else8 Reduce legit by 19 Copy the value of Aright to Aleft

10 Reduce right by 1

11 if Aleft=0 then reduce legit by 1.12 Stop

Exercising the Converging Pointers Algorithm

d1 d2 d3 d4 d5 d6 d7 d8

5 3 4 0 6 2 4 0legit

Measuring Efficiency by Counting Steps

The efficiency of an algorithm is the number of steps that it takes to complete its task. Sometimes, this is called the complexity of an algorithm.

Efficiency depends on the data. E.g., the search algorithm takes fewer steps to locate a value at the beginning of a list than to locate a value at the end of the list.

The “worst case” efficiency is the maximum number of steps that an algorithm can take for any collection of data values.

If the input has size n, efficiency will be a function of n