Practice Dependency Graphs in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A dependency graph is a directed graph where nodes are variables and arrows show which variables directly influence which others.

Like a flowchart: A affects B, B affects C. Arrows show dependencies.

Showing a random 20 of 50 problems.

Example 1

easy

In the chain

A \to B \to C \to D

, how many ancestors does

D

have?

Example 2

easy

A graph has edges

A \to B

A \to C

B \to D

. List the parents of

D

Example 3

medium

Find a valid evaluation order for

C = A + B

E = C + D

, where

A, B, D

are inputs.

Example 4

medium

Given

A \to B

A \to C

B \to D

C \to D

D \to E

, list one valid topological order.

Example 5

easy

In a dependency graph, the edge

X \to Y

means which variable directly influences the other?

Example 6

easy

Three tasks have the following dependencies: Task B depends on Task A, and Task C depends on Task B. Draw the dependency graph and determine a valid execution order.

Example 7

medium

Given

D = A + B

and

E = D \cdot C

, list every variable that

E

depends on, directly or indirectly.

Example 8

easy

A formula computes

C = A + B

. Draw the dependency arrows.

Example 9

medium

Variables: rainfall

\to

soil moisture

\to

crop yield, and temperature

\to

crop yield. If only temperature changes, does soil moisture change?

Example 10

easy

For

z = f(x) + g(y)

, draw the dependency graph and identify the leaf node.

Example 11

medium

In the graph

A \to B

A \to C

B \to D

C \to D

, what is the out-degree of

A

Example 12

medium

Tasks

A \to B

A \to C

B \to D

C \to D

. What is the minimum number of stages to complete all tasks if independent tasks can run in parallel?

Example 13

easy

Does correlation between

A

and

B

guarantee that one depends on the other? Answer yes or no.

Example 14

easy

A node with no incoming arrows in a dependency graph is called a ____ variable.

Example 15

medium

In a causal dependency graph

X \to Y \to Z

, intervening on

Y

(forcing its value) breaks which arrow?

Example 16

medium

Given the dependencies:

D

depends on

A

and

B

;

E

depends on

B

and

C

;

F

depends on

D

and

E

. Find all valid topological orderings.

Example 17

challenge

A graph claims

A \to B

B \to C

C \to A

is a valid computation pipeline. Identify the flaw and state the condition needed to fix it.

Example 18

medium

How many distinct topological orderings does the DAG

A \to B

A \to C

have? (No other edges.)

Example 19

medium

In the DAG with edges

A \to B

A \to C

B \to D

C \to D

D \to E

, what is the longest path length (in edges) from

A

E

Example 20

medium

In a spreadsheet, cell

C1 = A1 + B1

and

D1 = C1 \cdot 2

. If

A1

changes, which cells must be recomputed?

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Related Concepts

Functional Dependency