CSC165H1 Worksheet: Tutorial 8 Algorithm analysis (SOLUTIONS)

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1 CSC165H1, Witer 018 Learig Objectives By the ed of this worksheet, you will: Aalyse the ruig time of fuctios cotaiig ested loops. 1. Nested loop variatios. Each of the followig fuctios takes as iput a o-egative iteger ad performs at least oe loop. For each loop, determie the exact umber of iteratios that will occur (i terms of the size of the fuctio s iput), ad the use this to determie a Theta expressio for the ruig time of each fuctio. Whe there are ested loops, remember to do the followig: (i) First, determie a expressio for the exact cost of the ier loop for a fixed iteratio of the outer loop. This may or may be the same for each iteratio of the outer loop. (ii) The determie the total cost of the outer loop by addig up the costs of the ier loop from (i). Note that the if the cost of the ier loop is the same for each iteratio, you ca simply multiply this cost by the total umber of iteratios of the outer loop. Otherwise, you ll eed set up ad simplify a expressio ivolvig summatio otatio (Σ). (iii) Repeat steps (i)-(ii) if there is more tha oe level of estig, startig from the iermost loop ad workig your way outwards. Your fial result should deped oly o the iput size, ot ay loop couters. (a) 1 def f1(): i = 0 j = 0 5 while j < : 6 j = j i = 5 (b) For a fixed iteratio of the outer loop, the ier loop takes exactly iteratios, with each iteratio costig a sigle step. Note that this is the same for each iteratio of the outer loop. There are iteratios of the outer loop. We multiply with this by the cost of each outer loop iteratio 5 to get a total cost of, which is Θ( ). 5 1 def f(): i = j = 1 5 while j < : 6 j = j * 3 7 k = 0 8 while k < : 9 k = k + 10 i = 1 The first ier loop takes log 3 iteratios, with each iteratio costig a sigle step. The secod ier loop takes iteratios, with each iteratio also costig a sigle step. Agai, this is the same for each iteratio of the outer loop. There are max(, 0) iteratios of the outer loop, for a total cost of max(, 0) ( ) log 3 +, Page 1/5

2 CSC165H1, Witer 018 which is Θ( ). Note: if you wat, you ca simplify the cost of a sigle iteratio of the outer loop to be simply Θ(), sice the calculated cost is log 3 + Θ(). (c) 1 def f3(): i = 0 j = 5 while j > 0: 6 k = 0 7 while k < j: 8 k = k j = j i = (d) We start with the iermost loop (with k) for a fixed iteratio of the other loops. This executes exactly j times, with each iteratio costig a sigle step. Note that this cost depeds o the value of j. Next, we cosider the cost of the middle loop for a fixed iteratio of the outer loop. We eed to add up the cost of the iermost loop for each value of j from dow to 1. (Note that the summatio expressio will be the same as if j wet up from 1 to.) This total cost is j = j=1 Note that this cost does ot deped o the iteratio umber of the outermost loop. Fially, the outer loop rus times i total. This meas that the total cost is is Θ( 3 )., which Note: you ca look up a formula for sum of powers of or geometric series for the aalysis i this questio. This aalysis is trickier tha the others. 1 def f(): i = 1 j = 0 5 while j < i: 6 j = j i = i * The ier loop takes exactly i iteratios for a fixed iteratio of the outer loop, ad the cost per iteratio is 1 step. The outer loop rus log iteratios, where i takes o the values 0, 1,,..., log 1 before the loop termiates whe i = log 1. We have the followig summatio for the total ruig time of this fuctio: log 1 k=0 k = log 1 Page /5

3 CSC165H1, Witer 018 So the total cost is log 1. It turs out that log 1 Θ(), although we ll leave a proof of this as a exercise. Note that oe importat property of floor that s helpful here is that log 1 log < log + 1. Correctio: this used to read log iteratios. Page 3/5

4 CSC165H1, Witer 018. Cosider the followig algorithm: 1 def subsequece_sum(l): """L is a list of umbers.""" 3 = le(l) max = 0 5 for i i rage(): # Loop 1: i goes from 0 to -1 6 for j i rage(i, ): # Loop : j goes from i to -1 7 sum = 0 8 for k i rage(i, j + 1): # Loop 3: k goes from i to j 9 sum = sum + L[k] 10 if max < sum: 11 max = sum 1 retur max Determie the Theta boud o the ruig time of this fuctio (i terms of, the legth of the iput list). For practice, do ot make ay approximatios o the umber of iteratios of ay of the three loops; that is, your aalysis should actually calculate the total umber of iteratios of the iermost k-loop across all iteratios of the outer loop. Go slow! Treat this is as a valuable exercise i performig calculatios with summatio otatio. You may fid the followig formulas helpful (valid for all, a, b Z + ): i =, i = ( + 1), 6 b f(i) = i=a b a+1 i =1 f(i + a 1) As usual, we focus o the iermost loop (Loop 3). For a fixed iteratio of Loops 1 ad, the Loop 3 takes exactly j 1 iteratios, with a cost of 1 step per iteratio. So the cost of the iermost loop for a fixed iteratio of Loops 1 ad is j 1. Note that this umber of iteratios depeds o both i ad j. Now cosider the cost of all iteratios of Loop for a fixed iteratio of Loop 1. We eed to sum the cost of Loop 3 for all values of j from i to 1. The total cost for a fixed iteratio of the outer Loop 1 is: 1 j 1 = j=i = i j =1 i j =1 j (chage of variable j = j 1) j ( i)( 1) = = i = 1 i ( + 1 ) ( + ) [Note: we re groupig by i here because the ext step is to sum over possible values of i.] To obtai the total cost, we have to add up the cost of each iteratio of the outer i-loop (as i goes from 0 to Page /5

5 CSC165H1, Witer 018 1): 1 [ ( 1 i + 1 ) ( + )] = 1 1 ( i + 1 ) 1 ( + ) 1 1 = 1 ( ) ( ( 1)( 1) + 1 ) ( ) ( ( 1) ) ( 1)( 1) = 1 ( 1) = 3 = 3 3 = ( + 1)( 1) + + So the the total umber of basic operatios is , which is Θ(3 ). Page 5/5