The expense of constructing and maintaining bridges and the whole or such part as the council shall determine, of the expense of paving, improving, and working, including grading and graveling upon the streets, alleys, and highways, may be paid from the general city funds, to be raised by tax upon all the property in the City. The expense of grading, paving, or graveling or otherwise improving any street, alley, or highway, may be defrayed by a special assessment upon the lots and premises abutting upon such improvement, or a part of such expense may be so paid and the remainder may be paid from the general city funds as the council may decide. The lots and premises abutting upon a street, alley, or highway improvement shall constitute a special assessment district. The term “paving” shall include curbing and the construction of crosswalks in the paved streets. All assessments for street improvements shall be made upon the abutting lots and premises according to the number of feet frontage upon the improvement, and all assessments for alley improvements shall be made upon abutting lots and premises according to the benefits received by such lots and premises. When such assessment is to be made upon lots in proportion to their frontage upon the improvement, if from the shape or size of any lot an assessment thereon in proportion to the frontage would be unjust and disproportionate to the assessment upon other lots, the assessor making the assessments may assess such lot for such number of feet frontage as in his opinion will be just.