joinAll-steps.md

Step-by-Step Execution of Iterative joinAll for [ ["x", "y"], ["z"] ]

This document walks through the execution of the non-recursive iterative joinAll function for the input [ ["x", "y"], ["z"] ], which generates all possible permutations of strings from the input lists, joined by underscores. The function produces ["x_z", "y_z"]. Each step of the iterative process is detailed below, with a Mermaid diagram visualizing the state of the algorithm, including the result and new_result lists. The diagrams are compatible with GitHub’s Mermaid renderer.

Iterative joinAll Implementation

The iterative solution builds combinations by maintaining a list of partial combinations (result) and updating it for each sublist.

def joinAll(lls):
    result = [""]
    for sublist in lls:
        new_result = []
        for s in sublist:
            for r in result:
                if r == "":
                    new_result.append(s)
                else:
                    new_result.append(r + "_" + s)
        result = new_result
    return result

Input

lls = [ ["x", "y"], ["z"] ]
# Expected Output: ["x_z", "y_z"]

Execution Steps

Step 1: Initialization

• Description: Initialize result = [""] to start building combinations. This mimics the recursive base case, providing a starting point for combining strings.
• State:
  • result = [""]
  • new_result = [] (not yet used)
  • Current sublist: None
• Action: Set up the initial result list before processing the first sublist.

Mermaid Diagram: Initialization

graph TD
    A("Start joinAll") --> B("Initialize result = ['']")
    B --> C("result = ['']")
    C --> D("Proceed to first sublist")

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Step 2: Process First Sublist ["x", "y"] - Start

• Description: Begin processing the first sublist ["x", "y"]. Create new_result = [] to store new combinations. Start looping over the strings in the sublist (s = "x" first).
• State:
  • result = [""]
  • new_result = []
  • Current sublist: ["x", "y"]
  • Current string: s = "x"
• Action: For s = "x" and r = "" in result, since r is empty, append s to new_result.

Mermaid Diagram: Processing s = "x" in First Sublist

graph TD
    A("Start processing sublist ['x', 'y']") --> B("new_result = []")
    B --> C("s = 'x', r = ''")
    C --> D{Is r empty?}
    D -->|Yes| E("Append 'x' to new_result")
    E --> F("new_result = ['x']")

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Step 3: Process First Sublist ["x", "y"] - Continue

• Description: Continue processing the first sublist, now with s = "y". Combine s = "y" with r = "" in result, appending y to new_result since r is empty.
• State:
  • result = [""]
  • new_result = ["x"]
  • Current sublist: ["x", "y"]
  • Current string: s = "y"
• Action: Append y to new_result, resulting in new_result = ["x", "y"].

Mermaid Diagram: Processing s = "y" in First Sublist

graph TD
    A("Continue sublist ['x', 'y']") --> B("new_result = ['x']")
    B --> C("s = 'y', r = ''")
    C --> D{Is r empty?}
    D -->|Yes| E("Append 'y' to new_result")
    E --> F("new_result = ['x', 'y']")

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Step 4: Update Result After First Sublist

• Description: After processing all strings in ["x", "y"], update result = new_result. Move to the next sublist.
• State:
  • result = ["x", "y"]
  • new_result = [] (reset for next sublist)
  • Current sublist: None (moving to ["z"])
• Action: Set result = ["x", "y"] and prepare to process the second sublist.

Mermaid Diagram: Update Result

graph TD
    A("Finish sublist ['x', 'y']") --> B("new_result = ['x', 'y']")
    B --> C("Set result = new_result")
    C --> D("result = ['x', 'y']")
    D --> E("Proceed to sublist ['z']")

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Step 5: Process Second Sublist ["z"] - Start

• Description: Begin processing the second sublist ["z"]. Create new_result = []. For s = "z", combine with each r in result = ["x", "y"]. Since r is not empty, append r + "_" + s.
• State:
  • result = ["x", "y"]
  • new_result = []
  • Current sublist: ["z"]
  • Current string: s = "z", r = "x"
• Action: Append "x_z" to new_result.

Mermaid Diagram: Processing s = "z", r = "x" in Second Sublist

graph TD
    A("Start sublist ['z']") --> B("new_result = []")
    B --> C("s = 'z', r = 'x'")
    C --> D{Is r empty?}
    D -->|No| E("Append 'x_z' to new_result")
    E --> F("new_result = ['x_z']")

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Step 6: Process Second Sublist ["z"] - Continue

• Description: Continue processing s = "z" with the next r = "y" in result. Append "y_z" to new_result.
• State:
  • result = ["x", "y"]
  • new_result = ["x_z"]
  • Current sublist: ["z"]
  • Current string: s = "z", r = "y"
• Action: Append "y_z" to new_result, resulting in new_result = ["x_z", "y_z"].

Mermaid Diagram: Processing s = "z", r = "y" in Second Sublist

graph TD
    A("Continue sublist ['z']") --> B("new_result = ['x_z']")
    B --> C("s = 'z', r = 'y'")
    C --> D{Is r empty?}
    D -->|No| E("Append 'y_z' to new_result")
    E --> F("new_result = ['x_z', 'y_z']")

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Step 7: Final Update and Return

• Description: After processing the second sublist, update result = new_result. Since there are no more sublists, return result.
• State:
  • result = ["x_z", "y_z"]
  • new_result = []
  • Current sublist: None
• Action: Return ["x_z", "y_z"] as the final output.

Mermaid Diagram: Final Update and Return

graph TD
    A("Finish sublist ['z']") --> B("new_result = ['x_z', 'y_z']")
    B --> C("Set result = new_result")
    C --> D("result = ['x_z', 'y_z']")
    D --> E("Return result")

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Key Takeaways

• The iterative joinAll function builds combinations by updating a result list with new combinations for each sublist.
• Starting with [""] allows the algorithm to handle the first sublist correctly, mimicking the recursive base case.
• The underscore is added only when combining with non-empty strings, ensuring correct formatting.
• Each Mermaid diagram visualizes the state of result and new_result at each step, showing how combinations evolve.
• The process for [ ["x", "y"], ["z"] ] produces ["x_z", "y_z"] in seven steps: initialization, processing each string in each sublist, updating result, and returning the final output.