performance metrics and decorator patterns

Design Patterns for Data Structures

Performance Metricsand

Decorator Patterns

Chapter 4


Theoretical asymptotic bounds of sort algorithms4.3 Performance Metrics and Decorator Patterns 147

• Merge sort — Θ(n lgn)• Worst case quick sort — Θ(n2)

• Average case quick sort — Θ(n lgn)• Selection sort — Θ(n2)

• Insertion sort — Θ(n2)

• Heap sort — Θ(n lgn)

These bounds are all based on theoretical statement execution counts. In practice, theuser of any program is interested in the elapsed time for it to execute. To experimentallyverify the above theoretical asymptotic bounds you could try to time the execution withcalls to the system clock. The elapsed time is difficult to measure, however, becausea program runs under control of an operating system that typically manages scores ofprocesses simultaneously. Time measurements depend on external factors that are hardto control such as the number of other jobs in the system and the operating system’sscheduling policy.

The remainder of this chapter applies decorator design patterns to perform statementexecution counts experimentally without relying on the system clock. In software de-sign, a decoration is a functionality that is added to an existing operation to producesome desired side effect.

While it would be difficult to experimentally count the execution of every statementin a sorting program, a decorator pattern can count the execution of some parts of somestatements. The execution count is the side effect the pattern adds to the program. Be-cause the pattern does not provide a total count of the number of CPU cycles to do thesort, the count it does provide is a relative indication of the performance of the algorithm.

A metric is a standard of measurement. This section describes decorator patterns forthe following performance metrics.

• The number of array element comparisons.• The number of array element assignments.• The number of array element probes.

An example of an element comparison is in the comparison statementif (a[i] < a[child])

in the siftDown() method for heap sort. A decorator to count the number of compar-isons would increase the count by one for each execution of the if guard. A decoratorto count the number of probes would increase the count by two for each execution ofthe if guard, once for the access to a[i] and once for the access to a[child]. Anexample of an element assignment is the assignment statementa[mid] = a[j];

in the split() method of quick sort. A decorator to count the number of assignmentswould increase the count by one for each execution of the assignment statement.

This chapter describes two decorators that collect data using the above metrics.• Parametric decoration — for comparison and assignment metrics.• Decorator design pattern — for the probe metric.

Both decorator patterns use C++ templates in conjunction with function overloading.Strictly speaking, parametric decoration is not an OO design pattern as it contains noinheritance. Implementation of the probe metric uses the OO decorator design pattern.

Revised: August 30, 2016 Copyright ©: 1998, Dung X. Nguyen and J. Stanley Warford


How can we measure the performance?

Three possible metrics

Need a metric: a standard of measurement.

4.3 Performance Metrics and Decorator Patterns 147

















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One element assignment


Use decorator patterns to collect datausing these three metrics

4.3 Performance Metrics and Decorator Patterns 147

















Demo SortCompAsgn Project

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• Do not sort type double.• Instead, sort object with class CAMetrics.• In CAMetrics set up two counters, one to count the

number of comparisons and one to count thenumber of assignments.

• Overload the comparison operators to incrementthe comparison counter when executed.

• Overload the assignment operator to incrementthe assignment counter when executed.

Parametric decoration


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152 Chapter 4 Comparison Sort Algorithms

// ========= CAMetrics =========template<class T>class CAMetrics private:

static Counter _compareCount;static Counter _assignCount;

T _data;

public:CAMetrics(); // for array initialization.CAMetrics(CAMetrics const &rhs);T toT() const;void setT(const T &source);bool operator<(CAMetrics const &rhs);bool operator<(CAMetrics const &rhs) const;bool operator<=(CAMetrics const &rhs);bool operator<=(CAMetrics const &rhs) const;bool operator==(CAMetrics const &rhs);bool operator==(CAMetrics const &rhs) const;bool operator!=(CAMetrics const &rhs);bool operator!=(CAMetrics const &rhs) const;bool operator>=(CAMetrics const &rhs);bool operator>=(CAMetrics const &rhs) const;bool operator>(CAMetrics const &rhs);bool operator>(CAMetrics const &rhs) const;CAMetrics &operator=(CAMetrics const &rhs);

static int getCompareCount();static int getAssignCount();static void clearCompareCount();static void clearAssignCount();static void setCost(int cost);

public:friend ostream &operator<<(ostream &os, CAMetrics<T> const &rhs)

os << rhs._data;return os;

friend istream &operator>>(istream &is, CAMetrics<T> &rhs) rhs._assignCount.update();is >> rhs._data;return is;

;

Figure 4.31 CAMetrics.hpp. The CAMetrics class, which overloads the com-parison and assignment operators. The program listing continues in the next figure.

Copyright ©: 1998, Dung X. Nguyen and J. Stanley Warford Revised: August 30, 2016



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154 Chapter 4 Comparison Sort Algorithms

// ========= Comparison operators =========template<class T>bool CAMetrics<T>::operator<(CAMetrics<T> const &rhs)

_compareCount.update();return _data < rhs._data;

template<class T>bool CAMetrics<T>::operator<(CAMetrics<T> const &rhs) const

_compareCount.update();return _data < rhs._data;

template<class T>bool CAMetrics<T>::operator<=(CAMetrics<T> const &rhs)

_compareCount.update();return _data <= rhs._data;

template<class T>bool CAMetrics<T>::operator<=(CAMetrics<T> const &rhs) const

_compareCount.update();return _data <= rhs._data;

Figure 4.33 CAMetrics.hpp (continued). Implementation of the first two over-loaded comparison operators for the CAMetrics class. The other four operators aresimilar and are not shown. The program listing continues in the next figure.

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Copyright ©: 1998, Dung X. Nguyen and J. Stanley Warford Revised: August 30, 2016



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UHWXUQ BDVVLJQ&RXQWJHW&RXQW`

&OHDU PHWKRGV WHPSODWHFODVV 7!YRLG &$0HWULFV7!FOHDU&RPSDUH&RXQW ^

BFRPSDUH&RXQWFOHDU`

WHPSODWHFODVV 7!YRLG &$0HWULFV7!FOHDU$VVLJQ&RXQW ^

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VHW&RVW WHPSODWHFODVV 7!YRLG &$0HWULFV7!VHW&RVWLQW L&RVW ^

BFRPSDUH&RXQWVHW'HOWDL&RVW`

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WH[WVKRXOGFRQWDLQDOOOHWWHUVRIWKHDOSKDEHWDQGLWVKRXOGEHZULWWHQLQRIWKHRULJLQDOODQJXDJH 7KHUHLVQRQHHGIRUDVSHFLDOFRQWHQWV EXWWKHOHQJWKRIZRUGVVKRXOGPDWFKWRWKHODQJXDJH

+HOOR KHUHLVVRPHWH[WZLWKRXWDPHDQLQJ 7KLVWH[WVKRXOGVKRZ KRZDSULQWHGWH[WZLOOORRNOLNHDWWKLVSODFH ,I\RXUHDGWKLVWH[W \RXZLOOJHWQRLQIRUPDWLRQ 5HDOO\"



A Comparison Sort Bound

Any sort of n elements that is based on elementcomparison and exchange is (n lg n).


The proof uses the following lemma.


$ FRPSDULVRQVRUWERXQG2EYLRXVO\ WKHVRUWDOJRULWKPVWKDWDUHΘ(n lgn) DUHIDVWHUWKDQWKHRQHVWKDWDUHΘ(n2) $QDWXUDOTXHVWLRQLV &DQDQ\VRUWDOJRULWKPEHGHVLJQHGWKDWLVHYHQIDVWHUWKDQΘ(n lgn)"

$OOWKHVHVRUWVGHVFULEHGLQWKLVFKDSWHUDUHEDVHGRQSDLUZLVHFRPSDULVRQVRIYDOXHVLQDQDUUD\ $ UHPDUNDEOHWKHRUHPDERXWDOJRULWKPVWKDWGHSHQGRQVXFKSDLUZLVHFRPSDULVRQVLVWKHIROORZLQJ

7KHRUHP &RPSDULVRQVRUWERXQG $Q\VRUWRI n HOHPHQWVWKDWLVEDVHGRQHOHPHQWFRPSDULVRQDQGH[FKDQJHLV Ω(n lgn)

7KLVWKHRUHPDVVXPHVWKHFRPSDULVRQPHWULFDVDPHDVXUHRIWKHSHUIRUPDQFHRIWKHVRUWDOJRULWKP

%HFDXVH Ω LVDORZHUERXQG WKHWKHRUHPDVVHUWVWKDW QR FRPSDULVRQVRUWDOJRULWKPFDQEHEHWWHUWKDQΘ(n lgn) $FFRUGLQJWRWKHWKHRUHP WKHΘ(n lgn) VRUWVRIWKLVFKDSWHUDUHDEVROXWHO\WKHEHVWWKDWFDQEHDFKLHYHGE\DQ\RQH DQ\ZKHUH DQ\WLPH +RZFDQZHNQRZWKDWVRPHRQHLQWKHIXWXUHZLOOQRWEHFOHYHUHQRXJKWRLPSURYHRQWKHDV\PSWRWLFEHKDYLRURIWKHVHVRUWV" $IWHUDOO DQLQÀQLWHQXPEHURIDOJRULWKPGHVLJQVDUHSRVVLEOHDQGPDQ\KDYHQRW\HWEHHGGLVFRYHUHG

,QVSLWHRIWKHDSSDUHQWGLIÀFXOW\RISURYLQJWKHLPSRVVLELOLW\RIDSDUWLFXODUDOJRULWKPGHVLJQ WKLVVHFWLRQJLYHVVXFKDPDWKHPDWLFDOSURRI 7KHSURRIGHSHQGVRQWZRIDFWV³DOHPPDWKDWVSHFLÀHVDORZHUERXQGRQ n! DQGWKHELQDU\GHFLVLRQWUHHPRGHOIRUFRPSDULVRQVRUWDOJRULWKPV +HUHLVWKHOHPPDDQGLWVSURRI

/HPPD n! ≥ (n/2)n/2

3URRI

n!= ⟨'HÀQLWLRQRI n! DVVXPLQJ n LVHYHQ⟩

n · (n−1) · (n−2) · . . . · ( n2 +1) · ( n

2 ) · . . . ·3 ·2 ·1= ⟨0XOWLSOLFDWLRQLVV\PPHWULF a ·b = b ·a⟩

1n ·2(n−1) ·3(n−2) · . . . · ( n2 )(

n2 +1)

≥ ⟨3URGXFWRI n/2 WHUPV HDFKRIZKLFKLVJUHDWHUWKDQ n/2⟩(n/2)n/2

,WLVDQH[HUFLVHIRUWKHVWXGHQWWRVKRZWKDWLI n LVRGG WKHORZHUERXQGRQ n! LV(n/2)(n+1)/2 7KHFRPSDULVRQVRUWERXQGWKHRUHPLVVWLOOYDOLGZLWKWKLVERXQGRQ n!

7KHELQDU\GHFLVLRQWUHHPRGHOLVDWUHHUHSUHVHQWDWLRQRIWKHGHFLVLRQSURFHVVWKDWPXVWRFFXUZLWKDQ\FRPSDULVRQVRUW )RUH[DPSOH VXSSRVHWKHDOJRULWKPLVWRVRUWWKUHHLQSXWYDOXHV³ a b DQG c )RUWKHDOJRULWKPWREHFRUUHFW LWPXVWSURGXFHWKHFRUUHFWUHVXOWIRUDOOSRVVLEOHLQLWLDOSHUPXWDWLRQV 7KHUHDUHVL[SRVVLEOHSHUPXWDWLRQVRIWKUHHYDOXHV QDPHO\abc acb bac bca cab cba$ GHFLVLRQLQDFRPSDULVRQVRUWDOJRULWKPLVDQH[HFXWDEOHVWDWHPHQWWKDWLVXVXDOO\WKHJXDUGRIDQ LI VWDWHPHQWOLNHWKHDERYH LI VWDWHPHQWLQWKH VLIW'RZQ PHWKRGRIWKHKHDSVRUW ,QJHQHUDO WKHUHH[LVWVDPLQLPXPQXPEHURIVXFKFRPSDULVRQVWKDWPXVWEHPDGHLIWKHDOJRULWKPLVWREHFRUUHFWIRUDOOSRVVLEOHLQSXWSHUPXWDWLRQV

)LJXUHDVKRZVWKHGHFLVLRQWUHHIRUWKLVWKUHHLQSXWH[DPSOH 7KHQRWDWLRQa : b DWWKHURRWRIWKHGHFLVLRQWUHHUHSUHVHQWVWKHJXDUGRIDQ LI VWDWHPHQWRUDORRSWKDW



Height = 3

a : b

b : c a : c

a : c b : c

(a)!Maximum decisions = 3. Number of leaves = 6. (b)!Maximum decisions = 3. Number of leaves = 8.

≤ >

≤

≤ ≤

≤> >

>>

)LJXUH 7ZRGHFLVLRQWUHHV 3DUWDLVWKHGHFLVLRQWUHHIRUVRUWLQJWKUHHHOHPHQWV3DUWELVDIXOOWUHHFRQWDLQLQJWKHPD[LPXPQXPEHURIOHDYHVIRUDWUHHRIKHLJKW

FRPSDUHVYDOXHV a DQG b 7KHEUDQFKWRWKHOHIWLVIRUWKHSURFHVVLQJWKDWPXVWEHGRQHLI a ≤ b DQGWKHEUDQFKWRWKHULJKWLVWKHSURFHVVLQJWKDWPXVWEHGRQHLI a > b

7KHOHIWFKLOGRIWKHURRW b : c UHSUHVHQWVDFRPSDULVRQRI b DQG c ,WVOHIWFKLOGUHSUHVHQWVWKHRXWFRPHLI b ≤ c %HFDXVHWKHÀUVWWHVWGHWHUPLQHVWKDW a ≤ b DQGWKHVHFRQGWHVWGHWHUPLQHVWKDW b ≤ c WKHFRQFOXVLRQLVWKDWWKHVRUWHGRUGHULV ⟨a,b,c⟩ LQGLFDWHGE\WKHOHDI 2QWKHRWKHUKDQG LIWKHVHFRQGWHVWGHWHUPLQHVWKDW b > c WKHRQO\SRVVLEOHFRQFOXVLRQLVWKDW b KDVWKHPD[LPXPYDOXH 7RGHWHUPLQHZKHWKHUWKHFRUUHFWVRUWLV⟨a,c,b⟩ RU ⟨c,a,b⟩ UHTXLUHVWKHDGGLWLRQDOFRPSDULVRQRI a DQG c

,QJHQHUDO DQ\VRUWDOJRULWKPWKDWGHSHQGVRQSDLUZLVHFRPSDULVRQRIHOHPHQWVPXVWEHPRGHOHGE\DGHFLVLRQWUHHWKDWKDV n! OHDYHV EHFDXVHWKHUHDUH n! SRVVLEOHLQLWLDOSHUPXWDWLRQVRI n HOHPHQWVDQGWKHDOJRULWKPPXVWEHDEOHWRVRUWWKHHOHPHQWVFRUUHFWO\IRUDOOSRVVLEOHLQSXWSHUPXWDWLRQV 7KHUHIRUH WKHGHFLVLRQWUHHPXVWKDYH n! OHDYHV

)LJXUH VKRZVWKDWWKHKHLJKWRIWKHGHFLVLRQWUHHHTXDOVWKHPD[LPXPQXPEHURIFRPSDULVRQVWRVRUWWKHOLVW ,WFRXOGWDNHIHZHU ,QWKHDERYHH[DPSOH LI a ≤ b DQG b ≤ cWKHOLVWLVVRUWHGZLWKRQO\WZRFRPSDULVRQV %XW LI a ≤ b DQG b > c DWKLUGFRPSDULVRQLVUHTXLUHG

7KHΩ ERXQGH[LVWV EHFDXVHWRVRUW n HOHPHQWVWKHGHFLVLRQWUHHPXVWKDYH n! OHDYHV:KDWLVWKHVPDOOHVWKHLJKWDWUHHFDQKDYHLILWKDV n! OHDYHV" 3DUWERIWKHÀJXUHVVKRZVD SHUIHFW ELQDU\WUHHRIKHLJKWWKUHH $ SHUIHFWELQDU\WUHHRIKHLJKW h LVWKHWUHHZLWKWKHPD[LPXPQXPEHURIOHDYHV ,QJHQHUDO DKHLJKWh WUHHKDVDWPRVW 2h OHDYHV )RUH[DPSOH WKHKHLJKWWUHHLQ3DUWDKDVVL[OHDYHV DQGWKHKHLJKWWUHHLQ3DUWEKDVHLJKWOHDYHV ZKLFKLVWKHPD[LPXP

%HFDXVHWKHGHFLVLRQWUHHPXVWKDYH n! OHDYHV 2h ≥ n! +HUHLVDSURRIRIWKHFRPSDULVRQVRUWERXQGVWDUWLQJZLWKWKLVUHODWLRQVKLS

3URRI

2h ≥ n!= ⟨0DWK WDNHWKHORJEDVHRIERWKVLGHV⟩

h ≥ lg(n!)⇒ ⟨/HPPD n! ≥ (n/2)n/2 DQGPRQRWRQLFLW\RI lg⟩


23 68 6


Height = 3

a : b

b : c a : c

a : c b : c

(a)!Maximum decisions = 3. Number of leaves = 6. (b)!Maximum decisions = 3. Number of leaves = 8.

≤ >

≤

≤ ≤

≤> >

>>

)LJXUH 7ZRGHFLVLRQWUHHV 3DUWDLVWKHGHFLVLRQWUHHIRUVRUWLQJWKUHHHOHPHQWV3DUWELVDIXOOWUHHFRQWDLQLQJWKHPD[LPXPQXPEHURIOHDYHVIRUDWUHHRIKHLJKW

FRPSDUHVYDOXHV a DQG b 7KHEUDQFKWRWKHOHIWLVIRUWKHSURFHVVLQJWKDWPXVWEHGRQHLI a ≤ b DQGWKHEUDQFKWRWKHULJKWLVWKHSURFHVVLQJWKDWPXVWEHGRQHLI a > b

7KHOHIWFKLOGRIWKHURRW b : c UHSUHVHQWVDFRPSDULVRQRI b DQG c ,WVOHIWFKLOGUHSUHVHQWVWKHRXWFRPHLI b ≤ c %HFDXVHWKHÀUVWWHVWGHWHUPLQHVWKDW a ≤ b DQGWKHVHFRQGWHVWGHWHUPLQHVWKDW b ≤ c WKHFRQFOXVLRQLVWKDWWKHVRUWHGRUGHULV ⟨a,b,c⟩ LQGLFDWHGE\WKHOHDI 2QWKHRWKHUKDQG LIWKHVHFRQGWHVWGHWHUPLQHVWKDW b > c WKHRQO\SRVVLEOHFRQFOXVLRQLVWKDW b KDVWKHPD[LPXPYDOXH 7RGHWHUPLQHZKHWKHUWKHFRUUHFWVRUWLV⟨a,c,b⟩ RU ⟨c,a,b⟩ UHTXLUHVWKHDGGLWLRQDOFRPSDULVRQRI a DQG c

,QJHQHUDO DQ\VRUWDOJRULWKPWKDWGHSHQGVRQSDLUZLVHFRPSDULVRQRIHOHPHQWVPXVWEHPRGHOHGE\DGHFLVLRQWUHHWKDWKDV n! OHDYHV EHFDXVHWKHUHDUH n! SRVVLEOHLQLWLDOSHUPXWDWLRQVRI n HOHPHQWVDQGWKHDOJRULWKPPXVWEHDEOHWRVRUWWKHHOHPHQWVFRUUHFWO\IRUDOOSRVVLEOHLQSXWSHUPXWDWLRQV 7KHUHIRUH WKHGHFLVLRQWUHHPXVWKDYH n! OHDYHV

)LJXUH VKRZVWKDWWKHKHLJKWRIWKHGHFLVLRQWUHHHTXDOVWKHPD[LPXPQXPEHURIFRPSDULVRQVWRVRUWWKHOLVW ,WFRXOGWDNHIHZHU ,QWKHDERYHH[DPSOH LI a ≤ b DQG b ≤ cWKHOLVWLVVRUWHGZLWKRQO\WZRFRPSDULVRQV %XW LI a ≤ b DQG b > c DWKLUGFRPSDULVRQLVUHTXLUHG

7KHΩ ERXQGH[LVWV EHFDXVHWRVRUW n HOHPHQWVWKHGHFLVLRQWUHHPXVWKDYH n! OHDYHV:KDWLVWKHVPDOOHVWKHLJKWDWUHHFDQKDYHLILWKDV n! OHDYHV" 3DUWERIWKHÀJXUHVVKRZVD SHUIHFW ELQDU\WUHHRIKHLJKWWKUHH $ SHUIHFWELQDU\WUHHRIKHLJKW h LVWKHWUHHZLWKWKHPD[LPXPQXPEHURIOHDYHV ,QJHQHUDO DKHLJKWh WUHHKDVDWPRVW 2h OHDYHV )RUH[DPSOH WKHKHLJKWWUHHLQ3DUWDKDVVL[OHDYHV DQGWKHKHLJKWWUHHLQ3DUWEKDVHLJKWOHDYHV ZKLFKLVWKHPD[LPXP

%HFDXVHWKHGHFLVLRQWUHHPXVWKDYH n! OHDYHV 2h ≥ n! +HUHLVDSURRIRIWKHFRPSDULVRQVRUWERXQGVWDUWLQJZLWKWKLVUHODWLRQVKLS

3URRI




23 88 8


There are n! leaves in the decision treeto sort n elements.

A decision tree of height h has at most2h leaves.

2h n!

Design Patterns for Data Structures &KDSWHU &RPSDULVRQ6RUW$OJRULWKPV

3URRI



h ≥ lg(n/2)n/2

= ⟨0DWK lgab = b lga⟩h ≥ (n/2) lg(n/2)

= ⟨h = 7KHQXPEHURIGHFLVLRQV⟩7KHQXPEHURIGHFLVLRQV ≥ (n/2) lg(n/2)

= ⟨'HÀQLWLRQRI Ω⟩7KHQXPEHURIGHFLVLRQV = Ω((n/2) lg(n/2))

= ⟨3URSHUW\RI Ω GURSSLQJWKHFRQVWDQWV⟩7KHQXPEHURIGHFLVLRQV = Ω(n lgn)

7KHSURRIXVHVWKHFRUUHVSRQGHQFHEHWZHHQWKHQXPEHURIGHFLVLRQVLQWKHGHFLVLRQWUHHPRGHODQGWKHQXPEHURIFRPSDULVRQVLQWKHVRUWDOJRULWKP

&RPSDULVRQDQGDVVLJQPHQWPHWULFVZLWKSDUDPHWULFGHFRUDWLRQ>7REHZULWWHQ 7KHUHPDLQGHURIWKLVFKDSWHUFRQWDLQVSURMHFWFRGHIURPWKHGSGV'LVWULEXWLRQLQWHUVSHUVHGZLWKSDUDJUDSKVRIEOLQGWH[WDVÀOOHU@





performance metrics and decorator patterns

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