Evaluating Running Time of Algorithm

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Evaluating Running Time of Algorithm
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Principles of Running Time Analysis

• Measure time as a function of the input size
• Evaluate the worst-case performance for all
  inputs up to a given size
• Ignore constant factor
• Compare the functions that measure the time
  complexity ofalgorithms by their growth rate
• big-O for estimated upper bounds
• big- W for estimated lower bounds

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Running Time of Simple Statements

• Simple Statement
• arithmetic (, ,) operations
• logic operations (, .)
• comparisons (lt, )
• assignment (a b) with no function call
• read statement
• break, continue, return (simple control)

T(n) O(1)
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Running Time of Loops
for-loop
initialize
O(1)
test
O(1)
g(n) times
O(f(n))
body
establishing g(n) is usually simple.
O (g (n) f (n))
reinitialize
O(1)
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While Loops
while
O(1)
test
Need to establish a bound on the number of
iterations of the loop g(n). Might need an
inductive proof for g(n).
g(n) times
body
O(f(n))
O(g(n) f(n))
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While Loop - example
Searching an element with given value within an
array of size n i 0 (1) while ( x !
ai) (2) i (3)
(1) O(1) test in (2) O(1) (3)
O(1) iterations maximum n O(while-loop) O(1)
n O(1) O(n)
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Nested Loops
O (1)
initialize
test
O (1)
O (g (n) f (n))
g(n) times
inner-loop
O (f (n))
O (1)
reinitialize
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Nested Loop - example
Compute inside out
for i 1 to N for j 1 to N i i j
T(n) O (n2)
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Nested Loop - example
selectionSortI(int a, int n)
for (j 2, n, j) key Aj
k j-1 while (k gt 0 and Akgt key
Ak1 Ak k k-1
Ak1 key
T(n) ?
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Running Time of if-statement
test
O (max (f (n), g (n)))
if-part
else-part
O(f(n))
O(g(n))
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If Statement - example
if ( a1i 0) for (i 0, i lt n,
i) for (j 0, j lt n, j)
aij 0 else for ( 0, iltn,
i) aii 1
if T(n) O(n2) else T(n)
O(n) T(n) max (O(n2), O(n)) O (n2)
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Running Time of Consecutive Statements
O(f1(n))
O (f1 (n) f2 (n) f4 (n))
O(f2(n))
O (max (fi (n)), i 1, ., 4
O(f3(n))
O(f4(n))
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Consecutive Statements - example
for i 1 to n a1 0 for i 1 to N for
j 1 to N ai ai aj i j
T(n) O ( max (f (first-loop), f
(second-loops) O ( max (f (n), f
(n2)) O (f (n2))
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Recursive Rule for Bounding Running Time

• Recursive definition of a statement
• a simple statement (is a statement)
• Let S, S1 and S2 be statements, then the
  following are statements
• While (Cond) S
• for( E1, E2, E3) S
• If (Cond) S1 else S2
• If (Cond) S
• S1 S2 ...

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Example Insertion Sort
InsertionSortI(int a, int n)
for (j 2, n, j) 1 key Aj
2 k j-1 3
while (k gt 0 and Akgt key
Ak1 Ak 4 k k-1
5 Ak1 key 6
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Insertion sort - tree structure
Evaluate running time bottom-up
for
while
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6
2
4
leaves simple statements internal nodes
compound statements
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Programs with Function Call

• Non Recursive function calls
• Evaluated like compound statements. Bottom up.
• Recursive function calls
• Running time represented by a recursive function
• Requires techniques for solving recursive
  functions.