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Lecture13

# Lecture13 - CMPSC 24 Lecture 13 Recursion Divyakant Agrawal...

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Unformatted text preview: 5/12/10 CMPSC 24: Lecture 13 Recursion Divyakant Agrawal Department of Computer Science UC Santa Barbara 5/12/10 1 Lecture Plan •  Recursion –  General structure of recursive soluIons –  Why do recursive soluIons terminate? –  How do recursive programs manage the stack? –  Tail recursion –  When to use recursion? 2 What Is Recursion? Recursion like a set of Russian dolls. 3 1 5/12/10 What Is Recursion? •  Recursive call A method call in which the method being called is the same as the one making the call •  Direct recursion Recursion in which a method directly calls itself –  example •  Indirect recursion Recursion in which a chain of two or more method calls returns to the method that originated the chain –  example 4 Recursion •  You must be careful when using recursion. •  Recursive soluIons can be less eﬃcient than iteraIve soluIons. •  SIll, many problems lend themselves to simple, elegant, recursive soluIons. 5 Some DeﬁniIons •  Base case The case for which the soluIon can be stated non‐recursively •  General (recursive) case The case for which the soluIon is expressed in terms of a smaller version of itself •  Recursive algorithm A soluIon that is expressed in terms of (a) smaller instances of itself and (b) a base case 6 2 5/12/10 Finding a Recursive SoluIon •  Each successive recursive call should bring you closer to a situaIon in which the answer is known. •  A case for which the answer is known (and can be expressed without recursion) is called a base case. •  Each recursive algorithm must have at least one base case, as well as the general (recursive) case 7 General format for many recursive funcIons if (some condiIon for which answer is known) // base case soluIon statement else // general case recursive funcIon call 8 CompuIng Factorial Recursive deﬁni:on A deﬁniIon in which something is deﬁned in terms of a smaller version of itself What is 3 factorial? 9 3 5/12/10 CompuIng Factorial 10 Recursive ComputaIon 11 Factorial Program The func:on call Factorial(4) should have value 24, because that is 4 * 3 * 2 * 1 . For a situa:on in which the answer is known, the value of 0! is 1. So our base case could be along the lines of if ( number == 0 ) return 1; 12 4 5/12/10 Factorial Program Now for the general case . . . The value of Factorial(n) can be wriNen as n * the product of the numbers from (n ‐ 1) to 1, that is, n * (n ‐ 1) * . . . * 1 or, n * Factorial(n ‐ 1) And no:ce that the recursive call Factorial(n ‐ 1) gets us “closer” to the base case of Factorial(0). 13 Recursive Factorial int Factorial ( int number ) // Pre: number >= 0. { if ( number == 0) // base case return 1 ; else // general case return number * Factorial ( number - 1 ) ; } Why is this correct? 14 Three‐QuesIons for Verifying Recursive FuncIons •  Base‐Case Ques:on: Is there a non‐recursive way out of the func:on? •  Smaller‐Caller Ques:on: Does each recursive func:on call involve a smaller case of the original problem leading to the base case? •  General‐Case Ques:on: Assuming each recursive callworks correctly, does the whole func:on work correctly? 15 5 5/12/10 CompuIng ExponenIaIon Recursively •  From mathema:cs, we know that 20 = 1 and 25 = 2 * 24 •  In general, x0 = 1 and xn = x * xn‐1 for integer x, and integer n > 0. •  Here we are deﬁning xn recursively, in terms of xn‐1 16 // Recursive definition of power function int Power ( int x, int n) { if ( n == 0 ) return 1; // base case else // general case return ( x * Power ( x , n-1 ) ) ; } Can you compute mulIplicaIon recursively? How about addiIon? 17 Fibonacci Sequence Shall we try it again? Problem: Calculate Nth item in Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 • What is the next number? • What is the size of the problem? • Which case do you know the answer to? • Which case can you express as a smaller version of the size? 18 6 5/12/10 Fibonacci Program int Fibonacci(int n) { if (n == 0 || n == 1) return n; else return Fibonacci(n-2) + Fibonacci(n-1); ) That was easy, but it is not very eﬃcient. Why? 19 Recursive Linear Search struct ListType { int length ; // number of elements in the list int info[ MAX_ITEMS ] ; }; ListType list ; 20 Problem Instance PROTOTYPE bool ValueInList( ListType list , int value , int startIndex ) ; 74 36 . . . 95 list[0] [1] [startIndex] 75 29 47 . . . [length ‐1] index Already searched of Needs to be searched current element to examine 21 7 5/12/10 bool ValueInList ( ListType list , int value, int startIndex ) // Searches list for value between positions startIndex // and list.length-1 // Pre: list.info[ startIndex ] . . list.info[ list.length - 1 ] // contain values to be searched // Post: Function value = // ( value exists in list.info[ startIndex ] . . // list.info[ list.length - 1 ] ) { if ( list.info[startIndex] == value ) // one base case return true ; else if (startIndex == list.length -1 ) // another base case return false ; else // general case return ValueInList( list, value, startIndex + 1 ) ; } 22 22 “Why Use Recursion?” • Those examples could have been wrihen without recursion, using iteraIon instead. The iteraIve soluIon uses a loop, and the recursive soluIon uses an if statement. • However, for certain problems the recursive soluIon is the most natural soluIon. Build a prototype. A more eﬃcient iteraIve soluIon can be developed later. • Recursive soluIons are easier to reason about. • The FuncDonal Programming paradigm adopts recursion. 23 Printing List in Reverse struct NodeType { int info ; NodeType* next ; } class SortedType { public : . . . private : NodeType* listData ; }; 24 8 5/12/10 RevPrint(listData) listData A B C D E FIRST, print out this secIon of list, backwards THEN, print this element 25 Base Case and General Case Base case: list is empty Do nothing General case: list is non‐empty Extract the ﬁrst element; Print rest of the list (may be empty); Print the ﬁrst element 26 PrinIng in Reverse void RevPrint ( NodeType* listPtr ) { if { ( listPtr != NULL ) // general case RevPrint ( listPtr-> next ) ; //process the rest std::cout << listPtr->info << std::endl ; // print this element } // Base case : if the list is empty, do nothing } How would this work without recursion? 27 9 5/12/10 FuncIon BinarySearch( )   BinarySearch takes sorted array info, and two subscripts, fromLoc and toLoc, and item as arguments. It returns false if item is not found in the elements info[fromLoc…toLoc]. Otherwise, it returns true.   BinarySearch can be wrihen using iteraIon, or using recursion. 28 FuncIon BinarySearch( ) bool BinarySearch( ItemType info[ ], ItemType item, int fromLoc , int toLoc ) // Pre: info [ fromLoc . . toLoc ] sorted in ascending order // Post: Function value = ( item in info [ fromLoc .. toLoc] ) { int if mid; ( fromLoc > toLoc ) // base case -- not found return false ; else { mid = ( fromLoc + toLoc ) / 2 ; switch ( item.ComparedTo( info [ mid ] ) ) { case EQUAL: return true; //base case-- found at mid case LESS: return BinarySearch ( info, item, fromLoc, mid-1 ); case GREATER: return BinarySearch( info, item, mid + 1, toLoc ); } } Which version is easier: compare to iteraIve version presented next 29 IteraIve BinarySearch( ) bool BinarySearch( ItemType info, ItemType item, { int mid ; int first = fromLoc; int last = toLoc; bool found = false ; while (( first <= last ) && !found ) { mid = ( first + last ) / 2 ; switch ( item.ComparedTo( info [ mid ] ) ) { case LESS: last = mid - 1 ; break ; case GREATER: first = mid + 1 ; break ; case EQUAL: found = true ; break ; } } return found; } int fromLoc, int toLoc) 30 10 5/12/10 When a funcIon is called... •  A transfer of control occurs from the calling block to the code of the func:on. It is necessary that there be a return to the correct place in the calling block ader the func:on code is executed. This correct place is called the return address. •  When any func:on is called, the run‐:me stack is used. On this stack is placed an ac:va:on record (stack frame) for the func:on call. –  This stores all the variables local to the called func:on. 31 Stack AcIvaIon Frames •  The ac:va:on record stores the return address for this func:on call, and also the parameters, local variables, and the func:on’s return value. •  The ac:va:on record for a par:cular func:on call is popped oﬀ the run‐:me stack when the ﬁnal closing brace in the func:on code is reached, or when a return statement is reached in the func:on code. •  At this :me the func:on’s return value, if non‐void, is brought back to the calling block return address for use there. 32 Mystery Recursive FuncIon // Another recursive function int Func ( int a, int b ) { int result; if ( b == 0 ) result = 0; else if ( b > 0 ) result = a + // // Func ( a , b - 1 ) ) ; base case first general case // instruction 50 else // second general case result = Func ( - a , - b ) ; // instruction 70 return result; } 33 11 5/12/10 Run‐Time Stack AcIvaIon Records Run‐Time Stack AcIvaIon Records // original call is instrucIon 100 x = Func(5, 2); FCTVAL ? result ? b 2 a 5 Return Address 100 original call at instrucIon 100 pushes this record for Func(5,2) 34 Second Call x = Func(5, 2); // original call at instrucIon 100 FCTVAL ? result ? b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 call in Func(5,2) code at instrucIon 50 pushes on this record for Func(5,1) record for Func(5,2) 35 Run‐Time Stack AcIvaIon Records Third Call x = Func(5, 2); FCTVAL ? result ? b 0 a 5 Return Address 50 FCTVAL ? result 5+Func(5,0) = ? b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 // original call at instrucIon 100 call in Func(5,1) code at instrucIon 50 pushes this record for Func(5,0) record for Func(5,1) record for Func(5,2) 36 12 5/12/10 Run‐Time Stack AcIvaIon Records Third Call Completes x = Func(5, 2); FCTVAL 0 result 0 b 0 a 5 Return Address 50 FCTVAL ? result 5+Func(5,0) = ? b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 // original call at instrucIon 100 record for Func(5,0) is popped ﬁrst with its FCTVAL record for Func(5,1) record for Func(5,2) 37 Run‐Time Stack AcIvaIon Records Second Call Completes x = Func(5, 2); FCTVAL 5 result 5+Func(5,0) = 5+ 0 b 1 a 5 Return Address 50 FCTVAL ? result 5+Func(5,1) = ? b 2 a 5 Return Address 100 // original call at instrucIon 100 record for Func(5,1) is popped next with its FCTVAL record for Func(5,2) 38 Run‐Time Stack AcIvaIon Records First Call Completes x = Func(5, 2); // original call at line 100 FCTVAL 10 result 5+Func(5,1) = 5+5 b 2 a 5 Return Address 100 record for Func(5,2) is popped last with its FCTVAL 39 13 5/12/10 PracIce: Show AcIvaIon Records For These Calls x = Func( ‐ 5, ‐ 3 ); x = Func( 5, ‐ 3 ); What opera:on does Func(a, b) simulate? 40 Tail Recursion •  The case in which a func:on contains only a single recursive call and it is the last statement to be executed in the func:on. •  Tail recursion can be replaced by itera:on to remove recursion from the solu:on as in the next example. 41 Tail Recursion Example bool ValueInList ( ListType list , int value , int startIndex ) { if ( list.info[startIndex] == value ) // one base case return true ; else if (startIndex == list.length -1 ) // another base case return false ; else // general case return ValueInList( list, value, startIndex + 1 ) ; } 42 14 5/12/10 Equivalent IteraIve Version bool ValueInList ( ListType list , int value , int startIndex { while (list.info[startIndex] != value && startIndex != list.length-1 ) startIndex++ ; if ( value == list.info[ startIndex ] ) return true ; else return false; } ) So, what is the general logic? 43 Convert into IteraIve SoluIon int Power(int number, int exponent) { if (exponent == 0) return 1 else return number * Power(number, exponent - 1) } 44 IteraIve Equivalent int Power (int number, int exponent) { int val = 1; while (exponent != 0) { val = number*val; What is the logic? exponent‐‐; } return val; } 45 15 5/12/10 Tower of Hanoi Worked out on the blackboard 46 16 ...
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