Transcript Arrays, Records and Pointers

Arrays, Records and Pointers
Arrays, Records and Pointers
• Data structures are classified as either linear
or nonlinear.
• Normally arrays and linked lists fall into linear
data structures.
• Trees and graphs are treated as nonlinear
structures.
Linear Arrays
• A linear array is a list of a finite number n of
homogeneous data elements.
• The length of data elements of the array can
be obtained by the following formula:
• Length = UB – LB + 1
• UB is upper bound and LB is the lower bound.
Algorithm 4.1(Traversing a Linear
Array)
• Here LA is a linear array with lower bound LB and
upper bond UB. This algorithm traverses LA
applying an operation PROCESS to each element
of LA.
• 1. [Initialize counter] Set K := LB
• 2. Repeat steps 3 and 4 while K ≤ UB
• 3.
[Visit element] Apply PROCESS to LA[K]
• 4.
[Increase counter] Set K := K + 1
• 5. [End of Step 2 loop]
Algorithm 4.2(Inserting into a Linear
Array)
• INSERT(LA, N, K, ITEM)
• Here LA is a linear array with N elements and
K is a positive integer such that K ≤ N. This
algorithm inserts an element ITEM into the
Kth position in LA.
Algorithm 4.2(Inserting into a Linear
Array)
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1. [Initialize counter] Set J := N
2. Repeat steps 3 and 4 while J ≥ K
[Move Jth element downward]
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Set LA[J+1] := LA[J]
4.[Decrease counter] Set J := J – 1
[End of Step 2 loop]
5. [Insert element] Set LA[K] := ITEM
6. [Reset N] Set N := N + 1
7. Exit
Algorithm 4.3(Deleting from a Linear
Array)
• DELETE(LA, N, K, ITEM)
• Here LA is a linear array with N elements and
K is a positive integer such that K ≤ N. this
algorithm deletes the Kth element from LA.
Algorithm 4.3(Deleting from a Linear
Array)
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1. Set ITEM := LA[K]
2. Repeat steps for J = K to N - 1
[Move J + 1st element upward]
Set LA[J] := LA[J + 1]
[End of Step 2 loop]
3. [Reset the number N of elements in LA]
Set N := N - 1
4. Exit
Sorting; Bubble Sort
• Let A be a list of n numbers. Sorting A refers to
the operation of rearranging the elements of
A so they are in increasing order, i.e., so that
• A[1] < A[2]<………A[N]
How Bubble sort works?
• Suppose the list of numbers A[1], A[2],…..A[N]
is in memory.
• See page # 74 for detail.
Algorithm 4.4(Bubble Sort)
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BUBBLE (DATA, N)
1. Repeat steps 2 and 3 for K = 1 to N – 1
2. Set PTR := 1
3. Repeat while PTR ≤ N – K
(a) if DATA[PTR] > DATA[PTR + 1], then:
interchange DATA[PTR]and DATA[PTR + 1]
[End of if structure]
(b) Set PTR := PTR + 1
[End of inner loop]
[End of outer loop]
4. Exit
Complexity of Bubble Sort Algorithm
• 𝑓 𝑛 = 𝑛 − 1 + 𝑛 − 2 + ⋯…..2 + 1 =
𝑛(𝑛−1)
= O(n2)
2
Searching
• Linear Search
• In algorithm 2.4, we used the following while
loop header:
• Repeat while LOC ≤ N And DATA[LOC]≠ITEM
• Alternatively, we can do this as the first step:
• LOC = N + 1
• The purpose of this initial statement is to avoid
repeatedly testing whether or not we have
reached the end of the array DATA.
Algo 4.5(Linear Search)
• LINEAR(DATA, N, ITEM, LOC)
• Here DATA is a linear array with N elements
and ITEM is a given item of information. This
algorithm finds the location LOC of ITEM in
DATA, or sets LOC: = 0 if the search is
unsuccessful.
LINEAR(DATA, N, ITEM, LOC)
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1. [Insert ITEM at the end of DATA]
Set DATA[N + 1] = ITEM
2. [Initialize counter] Set LOC: = 1
3. [Search for item]
Repeat while DATA[LOC] ≠ ITEM
Set LOC := LOC + 1
[End of loop]
4. [Successful?] If LOC = N + 1, then Set LOC:=0
5. Exit
Complexity of Linear Search Algorithm
• Discuss it yourself (also see page # 77)
Binary Search
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Data is sorted in increasing order.
Extremely efficient algorithm.
This algorithm works as follows:
During each stage of our algorithm, our
search for ITEM is reduced to a segment of
elements of DATA:
• DATA[BEG], DATA[BEG + 1], DATA[BEG + 2]……DATA[END]
Binary Search
• Note that the variables BEG and END denote the
beginning and end locations of the segment
under consideration.
• Algorithm compares ITEM with the middle
element DATA[MID] of the segment, where MID
is obtained by:
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MID = INT((BEG + END)/2)
• If DATA[MID] = ITEM, then the search is successful
and we set LOC := MID, otherwise, a new
segment of DATA is obtained as follows:
Binary Search
• (a) If ITEM < DATA[MID], then ITEM can appear
only in the left half of the segment:
DATA[BEG], DATA[BEG + 1], DATA[BEG + 2]……DATA[MID - 1]
So we reset END := MID – 1 and begin searching
again.
• (b) If ITEM > DATA[MID], then ITEM can appear
only in the right half of the segment:
DATA[MID+1], DATA[MID + 2], ……DATA[END]
So we reset BEG := MID + 1 and begin searching
again.
Binary Search
• If ITEM is not in the DATA, then eventually we
obtain
• END < BEG
• This condition signals that the search is
unsuccessful and in such a case we assign
• LOC := NULL.
• Here NULL is a value that lies outside the set
of indices of DATA. (In most cases, we can
choose NULL = 0)
Algorithm 4.6 (Binary Search)
• BINARY (DATA, LB, UB, ITEM, LOC)
• Here DATA is a sorted array with lower bound
LB and upper bound UB, and ITEM is a given
item of information.
• The variables BEG, END and MID denote
respectively the beginning, end and the
middle locations of a segment of elements of
DATA. This algorithm finds the location LOC of
ITEM in DATA or sets LOC = NULL.
BINARY (DATA, LB, UB, ITEM, LOC)
• 1. Set BEG = LB, END = UB and
• MID = INT((BEG + END)/2)
• 2. Repeat steps 3 and 4 while BEG ≤ END and
DATA[MID] ≠ ITEM
• 3. if ITEM < DATA[MID], then
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Set END = MID – 1
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else
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Set BEG = MID + 1
• [End of if structure]
BINARY (DATA, LB, UB, ITEM, LOC)
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4. Set MID = INT ((BEG + END)/2)
[End of step 2 loop]
5. if DATA[MID] = ITEM, then:
Set LOC = MID
else
Set LOC = NULL
[End of if structure]
6. Exit
BINARY (DATA, LB, UB, ITEM, LOC)
• See example 4.9
• See yourself the complexity of the binary
search algorithm.
Multidimensional Arrays
• The arrays whose elements are accessed by
more than one subscript are termed as
multidimensional arrays.
• Two-Dimensional array
• A two dimensional m × n array A is a collection
of m.n data elements such that each element
is specified by a pair of integers (such as J, K)
called subscripts, with the property that
Multidimensional Arrays
• 1 ≤ J ≤ m and 1 ≤ K ≤ m
• The element of A with first subscript j and
second subscript k will be denoted by
• A[J, K]
Representation of Two-Dimensional
Arrays in Memory
• Let A be a two-dimensional m × n array.
• Although A is pictured as a rectangular array
of elements with m rows and n columns, the
array will be represented in memory by a
block of m.n sequential memory locations.
• The programming language stores the array A
• either (1) column by column, or what is called
column-major order
Representation of Two-Dimensional
Arrays in Memory
• Or (2) row by row, in row major order. See fig
4.10
• For a linear array LA, the computer does not
keep track of the address LOC(LA[K]) of every
element LA[K] of LA, but does keep track of
Base(LA), the address of first element of LA.
The computer uses the formula:
LOC (LA[K]) = Base(LA) + w(K - 1)
to find address of LA[K].
Representation of Two-Dimensional
Arrays in Memory
• Here w is the number of words per memory
cell for the array LA, and 1 is the lower bound
of the index set of LA.
• For Column-major order
• LOC(A[J, K]) = Base(A) + w[M(K-1) + (J-1)]
• For Row-major order
• LOC(A[J, K]) = Base(A) + w[N(J-1) + (K-1)]
• See example 4.12
Algorithm 4.7 (Matrix Multiplication)
• MATMUL(A, B, C, M, P, N)
• Let A be an M × P matrix array, and let B be a P
× N matrix array. This algorithm stores the
product of A and B in an M × N matrix array C.
MATMUL(A, B, C, M, P, N)
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1. Repeat steps 2 to 4 for I = 1 to M:
2. Repeat steps 3 and 4 for J = 1 to N:
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Set C[I, J] = 0
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Repeat for K = 1 to P:
C[I, J] = C[I, J] + A[I, K] * B[K, J]
[End of inner loop]
[End of step 2 middle loop]
[End of step 1 outer loop]
5. Exit
SPARSE MATRICES
• Matrices with relatively high proportion of
zero entries are called sparse matrices.