Problem Set Analysis Part

Chapter 5 Arrays and Generalized Tables

——"Data Structure Questions" - Yan Weimin. Wu Weimin Edition

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1. Basic knowledge questions

5.1❶ Assuming that there is a two-dimensional array A ₆_×₈ , each element is stored in adjacent 6 bytes, the memory is addressed by bytes, and the starting storage position (base address) of A is known to be 1000, calculate:

(1) The volume (ie storage) of array A;

(2) The address of the first byte of the last element a _{57 of array A;}

(3) When stored in rows, the address of the first byte of element a _{14 ;}

(4) The address of the first byte of element a _{47 when stored in columns.}

5.2❶ Assuming that the integer array A ₉_×₃_×_{5 × 8} is stored according to the low subscript priority , the byte address of the first element is 100, and each integer occupies four bytes. What is the storage address of the following elements?

(1) a₀₀₀₀ (2) a₁₁₁₁ (3) a₃₁₂₅ (4) a₈₂₄₇

5.3❶ List all elements a _{ijkl in the array A}₂_×₂_×₃_×₃ in order according to the high-subscript priority storage method (the rightmost subscript is the main order), in order to simplify the expression, you can only list (i, j, k, l) sequence.

5.4❶ Rewrite Equation (5-3) in Section 5.3.1 of the textbook into an equation form.

5.5❸ With an upper triangular matrix (a _ij ) _n_×_n , store its upper triangular elements in the array B[m] row by row (m is sufficiently large), so that B[k]=a _ij , and k=f ₁ (i)+f ₂ (j)+c. Try to derive functions f ₁ , f ₂ and constant c (requiring that f ₁ and f ₂ do not contain constant terms).

5.6❷ Given a tridiagonal matrix (a _ij ) _n_×_n , store the elements on its three diagonals in the array B[3][n], so that the element B[u][v]=a _ij , try Derive the formula for subscript transformation from (i, j) to (u, v).

5.7❸ Given a tridiagonal matrix (a _ij ) _n_×_n , store the elements of its three diagonals in the array B[3n-2] row by row, so that B[k]=a _ij , find:

(1) Use i, j to represent the subscript transformation formula of k;

(2) Use k to represent the subscript transformation formula of i and j.

5.8 ❸ Assume a quasi-diagonal matrix

Stored in the one-dimensional array B[4m] as follows:

Write the conversion formula for finding k from a pair of subscripts (i, j).

5.9❷已知A为稀疏矩阵，试从空间和时间角度比较采用两种不同的存储结构（二维数组和三元组表）完成求运算的优缺点。

5.10❷求下列广义表操作的结果：

(1) GetHead【(p, h, w)】;

(2) GetTail【(b, k, p, h)】;

(3) GetHead【((a, b), (c, d))】;

(4) GetTail【((a, b), (c, d))】;

(5) GetHead【GetTail【((a, b), (c, d))】】;

(6) GetTail【GetHead【((a, b), (c, d))】】;

(7) GetHead【GetTail【GetHead【((a, b), (c, d))】】】;

(8) GetTail【GetHead【GetTail【((a, b), (c, d))】】】.

注意：【】是函数的符号。

5.11❷利用广义表的GetHead和GetTail操作写出如上题的函数表达式，把原子banana分别从下列广义表中分离出来。

(1) L₁ = (apple, pear, banana, orange);

(2) L₂ = ((apple, pear), (banana, orange));

(3) L₃ = (((apple), (pear), (banana), (orange)));

(4) L₄ = (apple, (pear), ((banana)), (((orange))));

(5) L₅ = ((((apple))), ((pear)), (banana), orange);

(6) L₆ = ((((apple), pear), banana), orange);

(7) L₇ = (apple, (pear, (banana), orange));

5.12❷按教科书5.5节中图5.8所示结点结构，画出下列广义表的存储结构图，并求它的深度。

(1) ((()),a,((b,c),(),d),(((e))))

(2) ((((a),b)),(((),(d)),(e,f)))

5.13❷已知以下各图为广义表的存储结构图，其结点结构和5.12题相同。写出各图表示的广义表。

5.14❸已知等差数列的第一项为a1，公差为d，试写出该数列前n项的和S(n)(n≥1)的递归定义。

5.15❹写出求给定集合的幂集的递归定义。

5.16❸试利用C语言中的增量运算“++”和减量运算“--”写出两个非负整数a和b相加的递归定义。

5.17❸ 已知顺序表L含有n个整数，试分别以函数形式写出下列运算的递归算法：

(1) 求表中的最大整数；

(2) 求表中的最小整数；

(3) 求表中n个整数之和；

(4) 求表中n个整数之积；

(5) 求表中n个整数的平均值。

二、算法设计题

数组

5.18❺设计一个算法，将数组An中的元素A[0]至A[n-1]循环右移k位，并要求只用一个元素大小的附加存储，元素移动或交换次数为O(n)。

5.19❹若矩阵A_m_×_n中的某个元素a_ij是第i行中的最小值，同时又是第j列中的最大值，则称此元素为该矩阵中的一个马鞍点。假设二维数组存储矩阵A_m_×_n，试编写求出矩阵中所有马鞍点的算法，并分析你的算法在最坏情况下的时间复杂度。

5.20❺类似于以一维数组表示一元多项式，以m维数组：（a_j1j2_…_jm），0≤j_i≤n，i=1，2，…，m，表示m元多项式，数组元素a_e1e2_…_em表示多项式中x₁^e1x₂^e2…x_m^em的系数。例如，和二元多项式x²+3xy+4y²-x+2相应的二维数组为：

试编写一个算法将m维数组表示的m元多项式以常规表示的形式（按降幂顺序）输出。可将其中一项c_kx₁^e1x₂^e2…x_m^em印成c_kx₁Ee₁x₂Ee₂…x_mEe_m（其中m，c_k和e_j（j=1，2，…，m）印出它们具体的值），当c_k或e_j（j=1，2，…，m）为1时，c_k的值或“E”和e_j的值可省略不印。