Transcript Document 7893574

Efficiency of Algorithms
Csci 107
Lecture 5
• Last time
– Algorithms for
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Find all occurences of target
Find number of occurences of target
Find number of values larger than target
Find largest /smallest, sum, average
– Pattern matching
• Today
– Pattern matching algorithm
– Efficiency of algorithms
– Data cleanup algorithms
– Reading: start on Chapter 3, textbook
Pattern Matching
Problem: Suppose we have a gene (text) T = TCAGGCTAATCGTAGG and
a probe (pattern) P = TA. Design an algorithm that searches T to find the
position of every instance of P that appears in T.
– 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
• Algorithm:
– What is the 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
Pattern Matching
• Input
– Gene (text) of n characters T1, T2, …, Tn
– Probe (pattern) of m (m < n) characters P1, P2, …Pm
• Output:
– Location (index) of every occurrence of pattern within text
• 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
Comparing Algorithms
• Algorithm
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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 designing an 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
• (Time) 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
(Time) Efficiency of an algorithm
worst case efficiency
is the maximum number of steps that an algorithm can take for any
input data values.
best case efficiency
is the minimum number of steps that an algorithm can take for any
input data values.
average case efficiency
-the efficiency averaged on all possible inputs
- must assume a distribution of the input
- we normally assume uniform distribution (all keys are equally
probable)
If the input has size n, efficiency will be a function of n
Analysis of Sequential Search
• Time efficiency
– Best-case : 1 comparison
• target is found immediately
– Worst-case: 3n + 5 comparisons
• Target is not found
– Average-case: 3n/2+4 comparisons
• Target is found in the middle
• Space efficiency
– How much space is used in addition to the input?
Worst Case Efficiency for Sequential Search
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Get the value of target, n, and the list of n values
Set index to 1
Set found to false
Repeat steps 5-8 until found = true or index > n
5 if the value of listindex = target then
6 Output the index
7 Set found to true
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else Increment the index by 1
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if not found then
10 Print a message that target was not found
11 Stop
Total
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n
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3n+5
Order of Magnitude
• Worst-case of sequential search:
– 3n+5 comparisons
– Are these constants accurate? Can we ignore them?
• Simplification:
– ignore the constants, look only at the order of magnitude
– n, 0.5n, 2n, 4n, 3n+5, 2n+100, 0.1n+3 ….are all linear
– we say that their order of magnitude is n
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3n+5 is order of magnitude n:
3n+5 = (n)
2n +100 is order of magnitude n: 2n+100=(n)
0.1n+3 is order of magnitude n: 0.1n+3=(n)
….
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.
5
d1
3 4 0 6 2
d2 d3 d4 d5 d6
4 0
d7 d8
Legit
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)
Variables: n, A1, …, An, newposition, left, B1,…,Bn
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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)
Variables: n, A1,…,An, legit, left, right
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Get values for n and the list of n values A1, A2, …, An
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Set legit to n
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Set left to 1
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Set right to 2
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Repeat steps 6-14 until left > legit
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if Aleftt ≠ 0
7 Increase left by 1
8 Increase right by 1
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else
10 Reduce legit by 1
11 Repeat 12-13 until right > n
12 Copy Aight into Aright-1
13 Increase right by 1
14 Set right to left + 1
15 Stop
Exercising the Shuffle-Left Algorithm
5
d1
3 4 0 6 2
d2 d3 d4 d5 d6
4 0
d7 d8
legit
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)
Variables: n, A1,…, An, legit, left, right
1 Get values for n and the list of n values A1,
A2,…,An
2 Set legit to n
3 Set left to 1
4 Set right to n
5 Repeat steps 6-10 until left ≥ right
6 If the value of Aleft≠0 then increase left by 1
7 Else
8 Reduce legit by 1
9 Copy the value of Aright to Aleft
10 Reduce right by 1
11 if Aleft=0 then reduce legit by 1.
Exercising the Converging Pointers Algorithm
5
d1
3 4 0 6 2
d2 d3 d4 d5 d6
4 0
d7 d8
legit