TCS-FEEL: Topology-Optimized Federated Edge Learning with Client Selection

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. 2025 Oct 23;25(21):6534. doi: 10.3390/s25216534

Algorithm 1 Dijkstra-based upload routing in time instant t.

Require:
Directed graph $G^{(t)} = (V, E^{(t)}, ℓ^{(t)})$ with $ℓ^{(t)} (u, v) > 0$ ; UD index set $N = {1, \dots, N}$ ; BS index 0
Ensure:
For each $i \in N$ : shortest upload path $P_{i \to 0}^{(t)}$ and time $T_{↑}^{(t)} (i)$
1:
Build in-neighbor adjacency: for each $(x, y) \in E^{(t)}$ , append x to $InNbr (y)$ and store $ℓ^{(t)} (x, y)$
2:
Initialize: for all $v \in V$ , set $d (v) \leftarrow + \infty$ , $pred (v) \leftarrow ⌀$ ; set $d (0) \leftarrow 0$
3:
Initialize a min-priority queue Q keyed by $d (\cdot)$ and insert 0
4:
while Q not empty do
5:
$u \leftarrow Q . E x t r a c t M i n ()$
6:
for all $v \in InNbr (u)$ do
7:
$d^{'} \leftarrow d (u) + ℓ^{(t)} (v, u)$
8:
if $d^{'} < d (v)$ then
9:
$d (v) \leftarrow d^{'}$ , $pred (v) \leftarrow u$
10:
if $v \in Q$ then
11:
$Q . D e c r e a s e K e y (v)$
12:
else
13:
$Q . I n s e r t (v)$
14:
end if
15:
end if
16:
end for
17:
end while
18:
Output per UD
19:
for all $i \in N$ do
20:
if $d (i) = + \infty$ then
21:
$P_{i \to 0}^{(t)} \leftarrow []$ ; $T_{↑}^{(t)} (i) \leftarrow + \infty$
22:
else
23:
$path \leftarrow [i]$ , $v \leftarrow i$
24:
while $v \neq 0$ do
25:
$v \leftarrow pred (v)$ ; if $v = ⌀$ then $path \leftarrow []$ ; $d (i) \leftarrow + \infty$ ; break
26:
append v to $path$
27:
end while
28:
$P_{i \to 0}^{(t)} \leftarrow path$ ; $T_{↑}^{(t)} (i) \leftarrow d (i)$
29:
end if
30:
end for
31:
return ${(P_{i \to 0}^{(t)}, T_{↑}^{(t)} (i)) : i \in N}$