Explain the purpose of a given algorithm

Algorithm Design and Problem‑Solving (Cambridge IGCSE 0478)

Learning objectives

• Read and interpret a given algorithm (pseudocode, flowchart or structure diagram).
• Identify the five components of an algorithm: input, processing, output, validation, testing.
• State clearly the purpose of the algorithm and how it solves the problem.
• Relate the algorithm to the four stages of the Programme Development Life‑Cycle (PDLC).
• Apply relevant knowledge from the other syllabus blocks (data representation, transmission, hardware, software) when analysing a problem.

---

1. Programme Development Life‑Cycle (PDLC)

The Cambridge syllabus expects the following four stages for every programming task.

1. Analysis – understand the problem, define the purpose and list the required inputs/outputs.
2. Design – produce a clear algorithm (pseudocode, flowchart or structure diagram).
3. Coding – translate the design into a programming language.
4. Testing & Debugging – verify correctness with test data, use trace tables, and correct any faults.

Link to other syllabus blocks

Syllabus block	Relevance to algorithm design
Data representation	Choosing the correct numeric type (binary, two’s‑complement, floating‑point) influences input validation and memory usage.
Data transmission	If the algorithm processes data received from a network or USB, students must understand packets, simplex/half‑duplex, and error‑checking.
Hardware	CPU cycles, cache, and instruction set affect the efficiency discussion (time/space complexity).
Software	System software (OS, interrupts) and application software determine where the algorithm will run and any platform‑specific constraints.

---

2. Example problem – “Maximum value in a list”

2.1 Understanding the problem

Aspect	Description
Problem statement	Given a finite list of numbers, locate the greatest (maximum) number.
Input	An integer n ≥ 1 followed by n numeric elements L[1..n].
Output	The single value max such that max ≥ L[i] for every i = 1..n.

2.2 Decomposing the problem (Structure diagram)

┌─────────────────────────────┐
│   Maximum‑value problem      │
├─────────────────────────────┤
│ 1. Input handling            │
│ 2. Initialise max            │
│ 3. Comparison loop           │
│ 4. Output result             │
└─────────────────────────────┘

2.3 Standard methods catalogue (Cambridge)

Method	Typical use‑case	Example in these notes
Search (linear)	Find a particular item or an extreme value.	Maximum‑value algorithm.
Sort	Arrange data in a defined order.	Bubble sort – not shown.
Totalling	Calculate the sum of a series.	Summing exam scores – separate example.
Counting	Determine how many items satisfy a condition.	Counting occurrences of the maximum.
Minimum/Maximum	Locate the smallest or largest element.	Current algorithm (maximum).
Average	Compute the mean value.	Average of marks – separate example.

---

3. Algorithm specification (pseudocode)

Step	Action
1	Read integer n and the list L[1..n].
2	Validate input: If n < 1 → output “Invalid list size” and STOP. For each i check L[i] is numeric (and, if required, within a given range).
3	Set max ← L[1] (first element becomes provisional maximum).
4	FOR i ← 2 TO n DO IF L[i] > max THEN max ← L[i].
5	OUTPUT max as the largest value.

3.1 Validation fragment (pseudocode)

IF n < 1 THEN
    PRINT "Error: list must contain at least one element"
    STOP
END IF
FOR i ← 1 TO n DO
    IF NOT IS_NUMERIC(L[i]) THEN
        PRINT "Error: element", i, "is not a number"
        STOP
    END IF
END FOR

---

4. Explaining the purpose

The algorithm determines the maximum (largest) element in a finite collection of numeric data. This is a fundamental “search” operation used in many real‑world contexts, for example:

• Finding the highest exam score in a class.
• Identifying the greatest temperature recorded in a month.
• Selecting the most expensive product in an inventory list.

---

5. Why the algorithm works (correctness proof)

1. After step 3, max holds the greatest value among the first element (L[1]).
2. Each loop iteration examines the next element:
   • If the element is larger than the current max, max is updated.
   • Otherwise max remains unchanged.
3. When the loop terminates, every element L[i] (for i = 1..n) has been compared with max. Therefore max is the overall greatest value.

---

6. Efficiency (complexity analysis)

• Time complexity: each element is inspected exactly once → O(n).
• Space complexity: only a constant number of extra variables (max, i, n) → O(1).
• For the “max/min” problem O(n) is optimal because any algorithm must look at every element at least once to guarantee correctness.

---

7. Testing & Debugging (dry‑run & trace table)

Students should produce a trace table for at least two test cases – a normal case and an edge case (list length 1).

Test case	Input (n, L)	Step	i	L[i]	max (after step)
A (normal)	n = 5 L = [7, 3, 9, 2, 9]	Initialise	–	–	7
		Loop	2	3	7
		Loop	3	9	9
		Loop	4	2	9
		Loop	5	9	9
B (edge)	n = 1 L = [‑4]	Initialise	–	–	‑4
		Loop	– (none)	–	‑4

If the final max does not match the expected result, the trace will show where the logic failed.

---

8. Common misconceptions

• Skipping elements – every element must be examined; otherwise a larger value can be missed.
• Confusing “maximum” with “average” – the algorithm never adds or divides numbers; it only compares.
• Omitting the initial assignment – without a valid starting value the comparison step is undefined.
• Assuming the list is already sorted – the algorithm works for any order; sorting is unnecessary.

---

9. Extending the idea

• Minimum value – replace the comparison > with <.
• Index of the maximum – maintain an extra variable pos that records the index whenever max is updated.
• Counting occurrences of the maximum – after finding max, make a second pass to count how many times L[i] = max.
• Combined min‑max – keep both min and max in a single loop.

---

10. Suggested flowchart (exam‑style)

---

11. Quick reference for the other syllabus blocks

11.1 Data representation

Concept	Key points for exams
Binary ↔ Decimal	2ⁿ values, weight of each bit, e.g. 1011₂ = 11₁₀.
Hexadecimal	Group binary in 4‑bit blocks; 0‑9, A‑F. Example: 0x3F = 63₁₀.
Two’s‑complement	Negative numbers: invert bits + 1. Range for 8‑bit: –128 to +127.
Logical shift	Left shift = multiply by 2; right shift = integer division by 2 (discard remainder).

11.2 Data transmission

Aspect	Typical exam question
Packet structure	Identify header, payload, and trailer; calculate total bits.
USB vs Serial vs Parallel	Choose the most suitable method for a given data‑rate and distance.
Duplex modes	Explain why a full‑duplex link is required for simultaneous video & audio streaming.

11.3 Hardware

Component	Exam‑relevant facts
CPU (FDE cycle)	Fetch → Decode → Execute. Clock speed determines how many cycles per second.
Core & cache	More cores → parallelism; cache reduces memory‑access time.
Instruction set	RISC (few simple instructions) vs CISC (many complex instructions).
Embedded system	Dedicated hardware + software for a specific task (e.g., microwave controller).

11.4 Software

Category	Key points
System software	Operating system, device drivers, utilities; manages resources and provides interrupts.
Application software	Programs that solve user‑focused problems – e.g., a spreadsheet that uses the maximum‑value algorithm.
Programming languages	High‑level (Python, Java) vs low‑level (Assembly). Cambridge expects pseudocode/flowchart, not language‑specific syntax.

11.5 Problem solving & algorithmic techniques

• Identify input, processing, output.
• Choose the appropriate standard method (search, sort, totalling, counting, min/max, average).
• Design a clear structure diagram before writing pseudocode.
• Validate inputs – always check size and type.
• Test with at least three cases: normal, edge, and error condition.
• Analyse time and space complexity; justify why it is optimal.

---

12. Summary checklist for the exam

1. State the problem clearly and list all inputs/outputs.
2. Produce a structure diagram that shows the logical sub‑tasks.
3. Write correct pseudocode (or flowchart) with:
   • Initialisation of variables.
   • Proper loop bounds.
   • Validation steps.
4. Explain the purpose of the algorithm in one concise sentence.
5. Show a trace table for at least two test cases (including an edge case).
6. Give the time‑ and space‑complexity (use O() notation).
7. Link the solution back to the PDLC stage you are in.
8. Reference any relevant data‑representation or hardware facts if the problem mentions them.

Following this structure will satisfy the Cambridge IGCSE/AL assessment objectives (AO1–AO3) for algorithm design and problem solving.