Suppose that a client performs an intermixed sequence of (stack) push and pop operations. The push operations put the integers 0 through 9 in order onto the stack; the pop operations print out the return values. Which of the following sequence(s) could not occur?
The given sequence 4 3 2 1 0 9 8 7 6 5 is possible because:
• First, push 0, 1, 2, 3, and 4 into the stack (stack: 4 3 2 1 0) and then pop these integers 4, 3, 2, 1, and 0 from stack then the following sequence will be printed 4 3 2 1 0.
• Now, push 5, 6, 7, 8, and 9 into the stack (stack: 9 8 7 6 5) and then pop these integers 9, 8, 7, 6, and 5 from stack then the following sequence will be printed 9 8 7 6 5.
• Thus, the complete sequence will be 4 3 2 1 0 9 8 7 6 5.
Hence, the sequence in option (a) is possible.
The given sequence 4 6 8 7 5 3 2 9 0 1 is not possible because:
• In this sequence, starting integer is 4 that means 0 to 4 elements have been pushed. According starting integer only 4 is popped
• In the end of the sequence, 1 is popped after 0 which is not possible because 1 is pushed after 0. So, 1 has to be popped before 0.
Hence, the sequence in option (b) is not possible.
The given sequence 2 5 6 7 4 8 9 3 1 0 is possible because:
• First, push 0, 1, and 2 into the stack (stack: 2 1 0) and then pop only 2 from stack (stack: 1 0).
• Now, push 5 into the stack (stack: 5 1 0) and then pop only 5 from stack (1 0).
• Now, push 6 into the stack (stack: 6 1 0) and then pop only 6 from stack (1 0).
• Now, push 7 into the stack (stack: 7 1 0) and then pop only 7 from stack (1 0).
• Now, push 4 into the stack (stack: 4 1 0) and then pop only 4 from stack (1 0).
• Now, push 8 into the stack (stack: 8 1 0) and then pop only 8 from stack (1 0).
• Now, push 9 into the stack (stack: 9 1 0) and then pop only 9 from stack (1 0).
• Now, push 3 into the stack (stack: 3 1 0) and then pop only 3 from stack (1 0).
• Now, pop 1 and 0 from the stack.
• Thus, the complete sequence will be 2 5 6 7 4 8 9 3 1 0.
Hence, the sequence in option (c) is possible.
The given sequence 4 3 2 1 0 5 6 7 8 9 is possible because:
• First, push 0-4 into the stack (stack: 4 3 2 1 0) and then pop 4, 3, 2, 1, and 0 from stack.
• Now, push 5 into the stack (stack: 5) and then pop only 5 from stack.
• Now, push 6 into the stack (stack: 6) and then pop only 6 from stack.
• Now, push 7 into the stack (stack: 7) and then pop only 7 from stack.
• Now, push 8 into the stack (stack: 8) and then pop only 8 from stack.
• Now, push 9 into the stack (stack: 9) and then pop only 9 from stack.
• Thus, the complete sequence will be 4 3 2 1 0 5 6 7 8 9.
Hence, the sequence in option (d) is possible.
The given sequence 1 2 3 4 5 6 9 8 7 0 is possible because:
• First, push 0-1 into the stack (stack: 1 0) and then pop 1 from stack (stack: 0).
• Now, push 2 into the stack (stack: 2 0) and then pop only 2 from stack (stack: 0).
• Now, push 3 into the stack (stack: 3 0) and then pop only 3 from stack (stack: 0).
• Now, push 4 into the stack (stack: 4 0) and then pop only 4 from stack (stack: 0).
• Now, push 5 into the stack (stack: 5 0) and then pop only 5 from stack (stack: 0).
• Now, push 6 into the stack (stack: 6 0) and then pop only 6 from stack (stack: 0).
• Now, push 7, 8, and 9 into the stack (stack: 9 8 7 0) and then pop 9, 8, 7, and 0 from stack.
• Thus, the complete sequence will be 1 2 3 4 5 6 9 8 7 0.
Hence, the sequence in option (e) is possible.
The given sequence 0 4 6 5 3 8 1 7 2 9 is possible because:
• First, push 0 into the stack and then pop 0 from stack.
• Now, push 1, 2, 3, and 4 into the stack (stack: 4 3 2 1) and then pop only 4 from stack (stack: 3 2 1).
• Now, push 5 and 6 into the stack (stack: 6 5 3 2 1) and then pop 6, 5, and 3 from stack (stack: 2 1).
• Now, push 7 and 8 into the stack (stack: 8 7 2 1) and then pop only 8 from stack (stack: 7 2 1).
• Now, if 7 will be popped from the stack then the given sequence will not be possible.
Hence, the sequence in option (f) is not possible.
The given sequence 1 4 7 9 8 6 5 3 0 2 is not possible because:
• In the end of the sequence, 2 is popped after 0 which is not possible because 2 is pushed after 0. So, 2 has to be popped before 0.
Hence, the sequence in option (g) is not possible.
The given sequence 2 1 4 3 6 5 8 7 9 0 is possible because:
• First, push 0, 1, and 2 into the stack (stack: 2 1 0) and then pop 2 and 1 from stack (stack: 0).
• Now, push 3 and 4 into the stack (stack: 4 3 0) and then pop 4 and 3 from stack (stack: 0).
• Now, push 5 and 6 into the stack (stack: 6 5 0) and then pop 6 and 5 from stack (stack: 0).
• Now, push 7 and 8 into the stack (stack: 8 7 0) and then pop 8 and 7 from stack (stack: 0).
• Now, push 9 into the stack (stack: 9 0) and then pop 9 and 0 from stack.
• Thus, the complete sequence will be 2 1 4 3 6 5 8 7 9 0.
Hence, the sequence in option (h) is possible.
Hence, the sequences in option ( b ), ( f ), and ( g ) are not possible.