Finding how far it is from O'Neill to Bridgeport

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Heap:
   1 - O'Neill at t=0
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remove top and process
  O'Neill at t=0 leads to:
      Grand island at t=111 - new entry
  and Valentine at t=105 - new entry
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Heap:
   1 - Valentine at t=105
   2 - Grand island at t=111
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remove top and process
  Valentine at t=105 leads to
      O'Neill at t=210 - already processed
  and Chadron at t=242 - new entry
  and Thedford at t=171 - new entry
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Heap:
   1 - Grand island at t=111
   2 - Thedford at t=171
   3 - Chadron at t=242
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remove top and process
  Grand island at t=111 leads to:
      O'Neill at t=222 - already processed
  and Thedford at t=256 - already know better
  and North Platte at t=251 - new entry
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Heap:
   1 - Thedford at t=171
   2 - Chadron at t=242
   3 - North Platte at t=251
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remove top and process
  Thedford at t=171 leads to:
      Grand Island at t=316 - already know better
  and North Platte at t=235 - an improvement
  and Valentine at t=247 - already processed
  and Alliance at t=295 - new entry
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Heap:
   1 - North Platte at t=235
   2 - Chadron at t=242
   3 - Alliance at t=295
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remove top and process
  North Platte at t=235 leads to:
      Grand Island at t=375 - already proecssed
  and Thedford at t=299 - already processed
  and Ogallala at t=286 - new entry
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Heap:
   1 - Chadron at t=242
   2 - Ogallala at t=286
   3 - Alliance at t=295
__________________________
remove top and process
  Chadron at t=242 leads to:
      Valentine at t=379 - already processed
  and Alliance at t=296 - already know better
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Heap:
   1 - Ogallala at t=286
   2 - Alliance at t=295
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remove top and process
  Ogallala at t=286 leads to:
      North Platte at t=337 - already processed
  and Sidney at t=353 - new entry
  and Bridgeport at t=373 - new entry
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Heap:
   1 - Alliance at t=295
   2 - Sidney at t=353
   3 - Bridgeport at t=373
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remove top and process
  Alliance at t=295 leads to:
      Thedford at t=419 - already processed
  and Chadron at t=249 - already processed
  and Bridgeport at t=333 - an improvement
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Heap:
   1 - Bridgeport at t=333
   2 - Sidney at t=353
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remove top and process
  Bridgeport at t=333       This is our destination. Could stop now
                            but we'll continue
  leads to:
      Alliance at t=371 - already processed
  and Scotts Bluff at t=366 - new entry
  and Sidney at t=374 - already know better
  and Ogallala at t=420 - already know better
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Heap:
   1 - Sidney at t=353
   2 - Scotts Bluff at t=366
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remove top and process
  Sidney at t=353 leads to:
      Kimball at t=393 - new entry
  and Bridgeport at t=394 - already processed
  and Ogallala at t=420 - already processed
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Heap:
   1 - Scotts Bluff at t=366
   2 - Kimball at t=393
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remove top and process
  Scotts Bluff at t=366 leads to:
      Kimball at t=419 - already know better
  and Bridgeport at t=399 - already processed
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Heap:
   1 - Kimball at t=393
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remove top and process
  Kimball at t=393 leads to:
      Sidney at t=433 - already processed
  and Scotts Bluff at t=464 - already processed
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Heap:
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The heap is empty, so we have finished, and we know how
far every town in Nebraska is from O'Neill by the shortest
route.